File D 52.16 / 63
(AVIATION)
PUBLISHED BY THE CHIEF OF AIR SERVICE, WASHINGTON, D. C.
Vol. V December 1, 1924 No. 493
THE 'INVESTIGATION OF STRUCTURAL
MEMBERS UNDER COMBINED AXIAL
AND TRANSVERSE LOADS
SECTION I
(AIRPLANE SECTION REPORT)
Prepared by J. S. Newell
Engineering Division, Air Service
McCook Field, Dayton, Ohio
Augu¥ 7, 1924
WASHINGTON
GOVERNMENT PRINTING OFFlCE
1925
Ralph Brown D rau
LIBRARY on
HAY 7 7 liJt3
A NonDepoitory
uburn University
This report covers Parts I and II of the data resulting from an investigation of structural members subjected
to combined axial and transverse loads. It contains the theoretical portion of the development of the
United States Army Air Service formulas and includes tables of functions which greatly facilitate the use of
these formulas.
Studies are now being made as to the agreement of the deflections computed by the use of these formulas
with the deflections actually obtained by tests made on experimental struts and trusses, it being considered that
a satisfactory check on the depen ability of the formulas can be obtained in this way. When these studies,
which have now been carried far enough to indicate that the formulas are reliable, are completed, Section II of
this report will be issued.
The reason for publishing this report in two sections is to furnish the airplane industry immediately with a
more complete set of data on the development and use of the precise ·formulas than is now available. This
section is selfcontained in that it includes all information necessary to the intelligent use of the precise formulas
in the design of airplane structures.
(II)
CERTIFICATE: By direction of the Secretary of War the matter contained herein is published as. administrative
information and is required for tlre proper transaction of the public business.
THE INVESTIGATION OF STRUCTURAL MEMBERS UNDER
COMBINED AXIAL AND TRANSVERSE LOADS
PURPOSE OF THE INVESTIGATION
The purpose of the investigation, the results of which
are given in this report, was to obtain a satisfactory
method of computing the stresses in such members of
an airplane structure as are subjected to Pombincd
axial and transverse loads.
SUMMARY OF THE RESULTS OF THIS INVESTIGATION
The study of the various approximate methods, the
results of which are given in Part I, indicated that the
combination of the ordinary threemoment equation,
which does no~ provide for an axial load, with the
I
various approximate formulas gave results that would
SCOPE OF THE INVESTIGATION not be dependable on ~ structure_ such as an airplane
spar. As a result of tlus, the precise formulas of Berry
The investigation was initiated for the purpose ~f and MullerBreslau were investigated for the purpose
making a comparative stu_dy of a number of approx! uf reducing . the labor involved in their application.
mate methods for the design of members under com The formulas developed in Part II will be found to be
bined loads to ascertain their relative degrees of con somewhat less laborious than those of Berry and, with
servatism and ease of application. Several approxi the use of the special tables of sines, cosines, and
mate methods were studied and applied to a specific tangents, somewhat easier than MullerBreslau's
problem for purposes of comparison. A full discussion method. Formulas for loading conditions other than
of the methods and results obtained will be found in those given by Berry or Muller Breslau are also
Part I of this report. . developed in Part II, so that precise formulas are now
Although several of the approximate formulas were available for all loading co.nditions liable to occur in
found to agree quite well among themselves, th~re were airplane design.
some that gave results which were considerably The tests which are described in Part III were
different, so it was decided to investigate the precise sufficient to establish the theory upon which the
formulas developed by Mr. Arthur Berry in his paper, precise methods depend, but since they were made
"The Calculation of Stresses in Aeropla~e Wi_ng on small specimens under laboratory conditions they
Spars,"' and to compare the results thus obtamed with did not indicate how dependable the precise methods
those from the approximate methods. It was found would be under practical conditions. The tests which
that considerable discrepancy existed between the are discussed in Part IV were made on a small truss
results, the approximate methods being too conserva similar to the lift truss of an airplane and indicate that
t!ve in some cases and unsafe in others. for continuous members under combined load the
Attention Was then directed toward the precise precise formulas are more dependable than the approximethods
of Berry and of MullerBreslau 2 and an mate. The precise formulas gave results within .5
effort was made to simplify them and, if possible,_ to per cent of those indicated by the tests where the
reduce the purely mechanical part of the mathematical ordinarv methods were as much as 16 per cent off.
work inherent in their use. Part II is devoted to the The ~onclusions obtained from this investigation are
development of different forms of the precise formulas as follows:
for the different conditions of loading tha_t are_ en The precise formulas of MullerBreslau and Berry
countered in airplane structural design. It is entirely accurately represent the forces and stres:ses in members
theoretical in its treatment of members under com subjected to combined axial and lateral loads.
bined loading. The theoretical loading conditions for which these
Parts III and IV are devoted to tests made on spruce formulas are developed are sufficiently close to the
specimens subjected to combined loads for the purpose actual conditions that the formulas may be used
of checking the theory and the formulas developed in safely in practical design.
Part II of this report. The precise formulas developed in this investigation
Part III is concerned with simple, pinended struts. are fundamentally identical with the MullerBreslau
Part IV, with continuous members. and Berry formulas, but are easier to apply in prac
The appendix contains tables for use with the for tical design and cover more conditions of loading.
mulas developed in Part II and articles on subjects In fact, in many cases these precise formulas are
closely related to the formulas, their development and easier to apply than the best approximate formulas.
use. The approxima.te formulas studied, and they include
1 Trans. Royal Aeronautical Society, London, '1919.
2 Graphische Statik, Vol. II, part 2.
all those in common use, are too inaccurate for general
employment and may give very unsafe results.
(1)
The approximate formulas may be used in preliminary
design and in final design also if the secondary
stresses and slenderness ratio are small and the margin
of safety is large.
2
The precise formulas must be used in design if the
secondary stresses or slenderness ratio is ·1arge or the
margin of safety is small
THE ACTION OF A MEMBER UNDER COMBINED
AXIAL AND TRANSVERSE LOAD
A concise ·statement of the action of a strut under
combined loads is desirable to clarify the problem
entailed, as well as the methods for its solution which
have been investigated or developed in this report.
1' • Pt,UCT7PIY Vh/JE~ C<n81/Yell lMP.
f'• «Ptecnwrvl'fl)e,e "'re,e"' ""'" INfLll
It is apparent from the above discussion that an
axial compressive load, which increases the bending
moment at every point, is of far greater importance in
the design of members under combined loads than is
an axial tension which tends to decrease the bending.
For this reason the investigation has been confined
almost entirely to the case of a compressive load,
although some attention has been given to axial tension
and methods of providing for it.
It is apparent that the increments to the deflection
or bending moment should be represented by a series
of some sort which, if the axial load is not too great,
will converge, so that the limit of the series may be
taken to represent the condition when the member
comes to rest and is in equilibrium. If the member is
continuous over two or more supports, the moments
will be increased, both at the supports and in the spans,
by the application of the axial load. The ordinary
threemoment equation, which would be used on a
J.P~'!=::~!~i ~~~~~~::;:;;;'.:=;~;;;;;_;;;;_::;:_=:;:=~:;::;=;·~~~''P~ continuous beam in conj unction with the various ~ ;;';;;.,..,,,, ' ' ~] approximate methods for computing stresses under
 l combined loading, makes no provision for the change
Fro. I.
Figure 1 shows a beam acted upon by a side load of
w pounds per inch and an axial load of P pounds. Due
to the side load alone the ·beam deflects a distance y'
at a point x inches from the left support and the bend
m. g moment at t h at po.m t 1. s 2w x2  wLx. 2
If a compressive load, P, is applied as shown, the
moment at point x will be increased by Py', since the
load P acts at a distance y' from the axis of the deflected
member. This increase in moment causes a greater
deflection at x, which, in turn, causes a further increase
in the moment. If the load P is not great enough to
cause failure, these increments of the moment and
deflection will get smaller and smaller until the strut
comes into equilibrium and the deflection at x becomes
y. If, however, the load P is sufficiently large, the
·increments of deflection will be successively greater
and greater until the strut fails by buckling.
On the other hand, if P be tension instead of compression
the deflection y' due to the side load alone
will be reduced instead of incre~d and, as P is
increased, the strut will tend to straighten out and the
moment at any point will be reduced. The failure,
when it occurs, will be a tension failure and will not
be accompanied by buckling of the member.
in moments over the supports due to the axial load
and so vitiates the effect of any series or other devic~
used in the formulas to provide for the effects of the
axial load in the spans. This is an important point,
as it accounts for a large part of the discrepancy between
the approximate and the precise methods when
applied to continuous members, such as the wing spars
of an airplane.
The precise methods provide for the a]!:ial load both
in the threemoment equation and in· the formulas for
the moment in the spans by the use of mathematical
series. It so happens that the series used are identical
with those of the trigonometric functions, sines, cosines,
and tangents, but it should be borne in mind that they
are not connected with angles in any way. The same
results could be obtained by substituting the series for
the sines or cosines in the formulas, but since the limits
of these series have already been computed and tabulated
so that it'is far simpler to determine the value of
the sine from a table than to determine the limit of the
series it represents, the terms "sine" and "cosine" are
used in the precis.e formul~. Special tables have been
computed for sines, cosines, and tangents for a range of
the variable from 0 to 3.50. If desired, the variable
may be considered to be an angle expressed in radians,
and the sine, cosine, or tangent may be obtained from
anx set of trigonometric tables for this angle converted
to degrees and minutes.
PART ICOMPARISON OF APPROXIMATE METHODS
The six approximate methods investigated and
recorded in this report were studied by applying the
various formulas to one specific spar and loading.
The results are tabulated and compared and will be
found in detail in the following pages.
Figure 2 shows the member on which .the computations
were made. It is a section of a wing beam from
the Boeing GA2 airplane continuous over two spans.
It is subjected to a uniformly distributed transverse
load, axial loads, and a restraining moment at one
end caused by the continuity of the beam and a cantilever
overhang. The other end is pinned. The
moment at the intermediate support was found by the
ordinary threemoment equation which does not provide
for the effects of an axial load.
!'ft. StJJIJIJ
jected to combined axial and· lateral load is that given
on page 520, Volume II, of Johnson, Bryan, and
Turneaure's "Modern Framed Structures." For a
beam having hinged ends it is,
M Mo
mu·= P(L')2
l± 10 El
where Mo is the maxfmum moment with no axial load
and L' is. the distance between hinge points. The
negative sign in the denominator is used when the
axial load cause.s compression, the positive when it
causes tension.
This formula is developed for pinended members,
but can be applied to any member by considering the
~=0  f=UQ6:L_11::::::::::::::::;:::::::::&3~~~~~=?~~~;:;;;:;~~~~1SO
fJJ:2,~P!J! /IYCI/
~· '1.91111.2. lz,= 14.:J,Jff. 4 ~,,~ ?.~IJ11.Z.
l,=IJ8" oi...~l.j'Jw~
l'llfTe~/lll
l'f. fir e. c:
Sl'IVC~
I, 'PO, I()() iljttt~
FIG. 2
It will be .noted that the lateral load shown in Figure
2 acts upward in accordance with the conventions
used in airplane design. · All of the approximate formulas
have been derived for bridge or building structures
in which the loads act downward. This difference
in direction of loading causes considerable difficulty in
the matter of signs and great care is required when
employing the approximate formulas to use the.proper
signs.
The formula given in this report will be modified
where necessary to cpnform to the conventions· used in·
airplane structural design, which are as follows:
Forces are considered positive when acting upwards;
shear, when the algebraic sum of all forces acting on
the beam to the left of the section considered is positive;
bending moments, when they tend to cause compression
in the upper fibers of the part of the beam to
the right of the section; the slope of a line, when it
rises from left to right; and deflection, when the
deflected position of a point is above the original
position.
THE FIRST APPROXIMATE METHOD
sections beween points Qf inflecti'on as simple, pinended
spans. Where Mi and Mz, the restraining
moments at the points of support, have been determined,
the distance between the points of inflection
may be found, the axial load being negfocted, from
in which w is the lateral load per inch run and L
is the distance between . supports. · This expression is
applicable to a unifomify distributed load and similar
expressions can be developed for other loading eonditions.
The value of L' determined from this equation
is in error, due to the fact that the points of inflection
move somewhat under the action of the axial load,
which is neglected !lntirely in this expression. The
~ffect of this error is gJmerally not great, so that the
formula may be used for preliminary analyseR, etc,;
but its presence should be noted and borne in mind.
w(L')2 • M 0 may be found from 8 ' 1. e.; the moment
at midspan on a simple beam under a uniformly
distributed load.
One of the best known of the approximate formulas The application of this method to the beam of Figure
for computing the bending moment in a member sub 2 gives the following results:
(3)
For the 138inch span the distance between points
of inflection is
L'= 2 f(M2M1)2 V _M2+M1+V = 2 wL w 4
ic· 33,20050,300)2 33,200+50,300+1382= 79.2,, V 26X138 26 4
The maximum moment between points of inflection
will be
w(L' )2
M.= 8
26X79.22
8
20,450
11,065X (79.2)2
lOX 1,600,000X 14.39
Similarly, for the 93inch bay
 20,450 in. lbs.
 29,300 (11. lbs.
4
This method is sufficiently accurate for use in preliminary
analyses or designs of airplanes, but it should
not be depended upon for the final design of continuous
or restrained members under combined load.
Johnson 's formula, or either of the others, will give
.satisfactory results for pinended struts having a side
load and it. may safely be used for the design of such
members.
THE SECOND APPROXIMATE METHOD
A second wellknown formula for the maximum
moment in a pinended member under a combined
lateral and axial load is the socalled secant formula,
Mm ... = M ( L fP) T ·t· · · 0 ±Py sec 2 VE! · · he pos1 1ve sign is
· used whel! the axial load is compressive. This formula
may be applied to continuous members in the same
way that Johnson's formula was, i. e., by computing
the distance L' between the points of inflection and
. L' 2 v/ c033,200) 2+ 33,200 + 932 = 65 4 26 x 93 26 4 . · be . . w(L') 2
UL substituting for L. In this case Mo= 8
for a
tween points of inflection.
M,,__3= 26,200 in. lbs.
This formula is sometimes given as M01 ...
Mo
1~JL' )~
.,,.2 EI
or Mm ... =z1.:_~ where Q is the Euler load for the part
of the spar between the points of inflection. It will
be noted that the difference between this formula and
Johnson's is that ~ has been substituted for
1
1
0
in
the second term of the denominator. it similar formula
may be derived from the precise equation for the
maximum moment in a pinended member under combined
load, if the series represented by the sines and
cosines are substituted for the functions themselves.
When the resulti.ng expressi9n is simplified, neglecting
terms containing powers of E and I, it becomes
M ilf0 max.=W(L')i
l  48 EI 
which reduces to Johnson's formula on suustituting
1 5 ·
10 for 43· These last two formulas are applicable
only to the case where the axial load is compression.
Of the three formulas, the last is the most conservative,
although the difference between them is slight.
The maximum moments in the spans of a continuous
beam subjected to combined load are somewhat in
error when computed by this method. The greatest
source of error is the fact that the ordinary threemoment
equation does not provide for the effect of
the axial load, so that the computed moments at the
supports are less than they should be. This results
in the computed moments in the spans being incorrect,
since it affects the location of the points of inflection.
A second source of error is the fact that the points of
inflection move under the influence of the axial load,
an effect which is also neglected in these formulas.
 5w(L')•
uniformly distributed lateral load and y= 384 EI ·
For other conditions of loading the values of Mo and ?J
may be found similarly from the corresponding expressions
for the moment and deflection at the midpoint
of a beam of length L' under the given loading.
In this formula the primary deflection, y, is multiplied
by an infinite series, the limit of which is given
by the secant of the quantity, ~~· This is done to
obtain the deflection when the strut comes into equilibrium
under the combined load. This ultimate deflection
is then multiplied by the axial load P to obtain
the secondary moment and the latter quantity is
added to the primary bending moment to obtain the
total.
This ·formula is susceptible to the same criticism as
Johnson's when applied to continuous or restrained
members. It gives satisfactory results when applied
to pinended struts' with lateral loads,. but s?mew~at
more labor is involved in its use than Is required with
Johnson's formula.
When applied to the spar and loading shown in
Figur~ 2, M 0 , P, L', E, and I are the same as before.
For the 138inch span
5w (L')'
y= 384 EI
5X26X(79.2)' __ 0.579 in.
384X 1,600,000X 14.39
L' /P 79.2 _ / 11,065 O 87 Sec 0.87 2V EI 2 V 1,600,000X 14.39 .
= 
1= ~1 . (See tables in Appe11dix.)
cos 0.87 0.6448
11,065 (0.579)
M1 2 =  20,450+ 0.6448 20,450
9,900 =30, 350 in. lbs.
Similarly, for the 93inch span
M23= 27,850 in. lbs.
THE THIRD APPROXIMATE METHOD
A somewhat similar formula was used by the Forest
Products Laboratory in the development of a wing
beamfor the Navy type TB flying boat. The formula is
M max·= M. + 1.
2 1;1' where y is the primary deflec
1Q
tion due to the lateral load at the point of maximum
moment and Q is the allowable Euler load on the
section between pin points or between points of inflection
on a continuous or restrained beam. This
formula is applicable to members having an axial
compressive load and it · is open to . the same errors as
are formulas 1 and 2, which depend on the distance
between the points of inflection cpmputed without.
providing for the effect of the axial load.
M0 , P, .y, L', E, and I are the same as the values
used with formula 2, so that, when this formula is
applied to the beam shown in Figure 2, the results are:
For the 138inch span,
Q
=7r2EJ=7r2Xl,600,000X14.39
3620
l
(L')2 (79.2)2 , 0 bs.
M =M +1.2_l]/=Z0 450 +I.2Xll,065(0.579)
12 ° p ' ] 1 065
1 1' Q 36,200
= 20,45011,050= 31,500 in. lbs.
Similarly, for the· 93inch bay,
Mi3= 28,900 in. lbs.
THE FOURTa APPROXIMATE METHOD
On page 155 of "Flugzeugstatik," Van Gries gives an
approximate . formula for dete.rmining the maximum
moment in the span of a beam under combined loading.
0.16 wL2 .
This formula is Mmu. O.l6 PL2' where L1sonel
EI
half the span length. Van Gries derives this from the
socalled exact cosine formula for fixed ended beams.
It. is admittedly approximate and is probably none
too reliable. 'When applied to the beam shown in
Figure 2 this formula gives the following results:
For the 138inch bay
M _ Q.16X26X692 19,800
max. 0.16Xll,065X692 0.634
l 1,600,000X 14.39
= 31,200 in. lbs.
For the !!3inch bay
Mmu. = 14,300 in. lbs.
5
If the distances between points of inflection had been
used instead of the total lengths of the bays, the maximum
moments in the spans would have been only
 7,430 in. lbs. in the 138inch and  5,460 in. lbs. in
the 93inch bay. These results do not agree with
those obtained by any of the other methods and are
probably much too small.
It is not recommended.that this formula be used for
the design of airplane members subjected to combined
loads, as it is believed that the good results on the
first example above are purely accidental.
THE FIFTH APPROXIMATE METHOD
An approximate method that has been much used.in
airplane design is that shown in Chapter III of "Structural
Analysis and Design of Airplanes." Briefly, this
method consists in assuming that the maximum total
moment occurs at the point of zero shear, computing
the deflection of the point of zero shear due to the
lateral load only, multiplying t.his deflection by the
axial load, and adding the result to the moment at that
point due to the lateral load alone. This method is
in error for three. reasons: It depends on values for the
moments at the supports which are computed without
considering the effect of the axial load; the point of
maximum total moment and the point of maximum
moment due to side load only are assumed to be iden·
tical; and only the first of the infinite series of secondary
moments is considered.
Applying this method to the 138inch bay, the results
are as follows:
M2M1 wL
Shear at left end, S+1 L z
33,20050,300 26Xl38
138 2 1,918 lbs.
Location of point of zero shear from left end,
X= 1,918/26=73.8 in.
Max. moment due to side load only
wx2
M x= M, + (S+1)x+z=50,3001,918X 73.8
+13X73.82=20,450 in. lbs.
Deflection at point of zero shear
y= ;1 cxL{~1 +8;1 cx+L)+~(xz+xL+V) J
= 73.8(73.8~138.0)[5o,3oo _:_ 1,918 <73 8+ 138 O)
1,600,000X14.39 2 6 · ·
+;~(73.82 +73.8X 138.+ 1382]
== + 1.025 in .
Van Gries questions the dependability of this for Secondary moment Py= 11,065X1.025= 11,350
mula, but it appears to give fairlygood results in this in. lbs.
case. The maximum moment in the long bay agrees Total moment Mmax.=Mx+Py=20,45011,350
quite closely.with that obtained by the other approxi = 31,800 in. lbs.
mate formulas, while that in the shorter span is in Applying the same method to the 93inch bay, the
fairly good agreement with the results of the precise maximum bending moment is 22,520 in. lbs.
methods. No provision is made for different degrees The value of the maximum deflection might be used
of restraint at the supports of a continuous beam, since in: computing the. secondary moment instea.d of the
none of the terms in this formula depend on these deflection at the point of zero shear, but the difference
moments or upon the points of inflection computed in the result will seldom be great, and the <:omputafrom
them. tions required are very tedious,
6
THE SIXTH APPROXIMATE ME'tHOD
The United States Army Air Service has sometimes
used a method called that of "secondary deflections"
for computing the moments in airplane members subjected
to combined axial and transverse loads. The
method is outlined on pages 67 and 68 of the 1920
editfon of. "Structural Analysis and the Design of
Airplanes." In this method, the axial load being
neglected, the bending moment, M0 , and deflection,
y., are computed at the point of zero shear in the span
under consideration.. The deflection thus obtained
w.ben multit>lied by the axial load gives a secondary
moment, Py.. The magnitude of the uniformly dis ·
tributed load over the entire span, wliich would give
a moment equal ·to Py0 is then computed, and also
the ·deflection due to such a load. The ratio of this
secondary deflection to the primary is used as the
constjl.nt, r, in a geometric series where the . ultimate
deflection, when the member· comes into equilibrium,
is y=y0 (l+r+r2+ri+ rn1). The total moment
is then M = M 0 + Py or M = M 0 + M' (1 +r+r2+ri+' __ ),
whi. ch becomes M = M· M' 0+· 1.00r
This method was applied to the beam of Figure 2
with the following results. As far as possible the
values obtained by the fifth method were used.
In the 138inch bay the primary moment at the
point of zero shear is 20,450 in. lbs.
The primary deflection of this point, y 0 = 1.025 in.
The first secondary moment is Py0 = 11,350 ip. lbs.
w', the uniformly distributed load that would give
this moment at the center of a simple supported beam
138 inch long is
I . BM 8:X 11,350
w = v= 138 X
138
4. 77 lbs. per in.
r= 42~7 =0.1834.
The maximum moment in the 138inch span then
becomes
M=M +  ¥:..___ · 11,350
0 .1.00r 2o,45o+1.ooo.1834
= 20,45013,900= 34,350 in. lbs.
For the 93inch bay this method gives a value of
M= 26,270 in. lbs ..
This method is probably as reliable as any of the
approximate fonp.ulas, but it, too, is dependent upon
the moments at the points of support computed from
the ordinary threemoment equation, which does not
provide for the axial load. For this reason the
assumed point of maximum moment, the point of zero
shear, is not correctly located, so that the maximum
moment as obtained by this method is in error both
as to magnitude and location.
THE PRECISE METHOD
For purposes of comparison the moments at the
strut points and in the spans were computed by the
formulas developed in Part II of 'this report.
The first step is to determine the moments at the
points of support when the effect of the axial load is
provided for. The form of the threemoment equation
for this case· is .
a, M1 L,+2Ms (fl1 Lt+fl2 L2)+a2 Ma L2
w, L11 +w2 L2
3
=4 'Yi 4 'Y2
. Substituting the proper values for a, fl, y, L, etc.,
M2=46,100 in· lbs.
The values for the maximum moments il1 ·the bays
are 43,400 in. lbs. for the 138in.ch span and 8,900
in. lbs. for the 93inch span.
The method of obtaining these values is given in
Part II of this report.
WEBB AND THORNE'S METHOD
A method which is similar to this precise method
in its derivation but is different from it in that approximate
algebraic coefficients are used in place of
the exact trigonometric functions ·occurring in the
precise formulas is to be found on page 121 et seq. of
Pippard and Pritchard's "Aeroplane Structures."
This method is accredited to Messrs. Webb and
Thorne, and it gives results that are in close agreement
with those obtained by the precise formulas so long
as the ratio of P/Q is less than about 0.85. This
method permits the computation of the strut point
moments by a threemoment equation which provides
for the effect of axial loads. The formulas for the
maximum moments in the bays also provide for such
loads. Webb and Thorne's method has an advantage
in that it requires no tables, but the computations
involved in its use are somewhat greater than those
necessary with the precise formulas given in Part II
when the tables are at hand. It is stat~d that this
method will give the strut point m~ments within
onehalf of 1 per cent when the ratio of P/Q is less
than 0.83, while the probable error in the maximum
moment in a span is less than 5 per cent under the
same conditions.
The Webb and Thorne equation of three moments
for the case of a uniformly distributed side load and
an axial compression is
(Qi~i) L,{ Mi(1+0.2 ~:}+2M2 (1  0.38 ~:)
i w, L1
2(10.014 ~:)}
+ (Q2~2) LJ Ma( l+0.2~;)+2M2(10.38 ~:)
i W2L2~(10.014 ~:) }=0
where Pis the axial load on the spar;
,,.2 EI
Q=IF;
w=the uniform load, acting upwar<l;
L=the span length.
\
The expression for the maximum moment in a span
is found from
Mmax.=Q~P{ (M,~M2) (1+0.26 ~)1.028 wL2}
(M2M1)2
 · 2wL2
The distance from the lefthand support to the poiut
of maximum moment is
X=!::_J.:f~~~J
, 2 wL
!l
Application of this method to the structure used
for purposes of comparison gives the following results.
The moment at the intermediate strut point, M2,
is found to be 46,100 in. lbs., while the maximum
moment in the 138inch bay is 45,620 in. lbs. For
the 93inch bay the maximum moment is given as
3,570 in. lbs., which is not in very good agreement
with the results from the precise equations, while the
other two values given above are a very close accord.
The ratios of P /Q are high in both of these bays and
the results depend on relatively small differences of
large numbers, which probably accounts for the differences
found. This is particularly true in the case of
the shorter span where the difference between two
quantities is so,· small that a slight error in the com
11utations will c,!rnnge the sign of the result from positlve
to negative.
COMPARISON OF THE VARIOUS RESULTS
Table I gives the values of the moments at the points
of support and in the spans of the beam shown in
Figure 2 as they were computed by the various methods
discussed in the foregoing pages. The value of the
moment at the outer support is constant in each
method, as it depends on the load on the cantilever tip.
The inner end of the beam is pinned to the fuselage,
so that M3 is assumed to be zero in each case. The
moment at the intermediate support, M2, is obtained
from the threemoment equation and the moments
in the spans by one of the formulas discussed above.
TABLE I
7
One of the sources of error involved in the use of all
of the approximate formulas is the fact that the
ordinary threemoment equation which is used in
the computation of the ·moments at the supports does
not provide for the effect of the axial load. It would
therefore appear that this error could be eliminated
if the precise threemoment equation were used. So
far as the moments at the points of support are concerned
·this is true, but a little study of the methods
outlined above will show that the moments in the
spans would still be in error even if the computations
were modified and made to depend on the moments
obtained by the precise threemoment equation. The
approximate formulas which ·depend on the distance
between points of inflectionwould give values of the
maximum moments in the spans that are less than
those already computed, as the effect · of the greater
moment at the intermediate support would be to
lessen the distance betwern the points of inflection.
This would be undesirable in the long span of the
illustrative beam, as the maxima computed by the
approximate formulas are ·already on the unsafe side
in compariso1i with the value determined from the
precise method.
The other approximate formulas would be similarly
affected; the larger moment at the intermediate support
as computed by the precise method would give smaller
values of the maximum moments in the spans.
RESULTS OF THE STUDY OF THE APPROXIMATE
FORMULAS
As a result of this study of various types of formulas
for providing for the effect of an axial load in combination
with a lateral load on a spar, it has been concluded
that none of the approximate formulas are satisfactory
for general use in the final design of a continuous or restrained
spar. Jo_hnson's formula is easy to apply and,
as it gives reliable results for pinended struts with a
lateral load, it may be used in the final design of such
members.
It has been seen that the principal source of error in
applying the approximate formulas to continuous members
arises from the use of the ordinary threemoment
equation for computing the moments at the points of
support. The inoments so obtained are in error, as no
provision for the effect of the axial load is made in their
Method I J,f,_, i M ,
1_ ____________ ______ +50,300 29, 300 +33,200 '1  26,200 I o
2 ..            +so, 300 30, 350 +33, 200  27, 850 o
3 ___ ______ _____ _____ +50, 300 31, 500 . +33, 200 28, 900 0
4 _________ __________ +50,300 31,200 +33,200 14,300 0
5 _______ ___ ____ _____ +50, 300 31, 800 +33, 200 22, 520 0
6.      +.'iO, 300 34 350 +33, 200 26, 270 0
Precise ____ ___ ______ +o0,300 43;400 +46, 100 8,900 o
Webb & Thorne .. .. +50,300 45,620 +46, 100 ~, 570 o
j computation. Attention has also been called to the
fact that the use of the values of the moments at the
supports, computed by the precise threemoment equation,
in conjunction with the approximate formulas is
lia.ble to increase the difference between the computed
and the actual maximum moments in the spans. Moreov.
er, once the precise threemoment equation has been
solved to determine the moments at the supports, it is
A comparison of the values obtained from the various
approximate formulas shows that they agree quite
well amongst" themselves, but when they are compared
to the moments computed by the precise equations,
which provide for the effect of the axial load, the
approximate methods show up as unsafe in some places
aud conservative in others.
2222325t2
far less laborious to determine the maximum moments
in the spans by the precise method than by the approximate
ones.
Webb & Thorne's method, which is really a modified
form of the precise equations that is approximate because
of the fact that the limit of the series introdured
bv the secondarv stresses is expressed as an algebraic
c~efficient inste~d of a trigonometric funetion, will give
satisfactory results when the axial load is not too close
to the Euler load. This method will usually give satisfactory
results for use in the design of airplane wing
spars, but its use will in general entail more arithmetical
work than the precise methods.
8
The final design of continuous or restrained members
subjected to combined loading should never be made on
the basis of the results obtained from the ordinary threemoment
equation in conjunction with any of the approximate
formulas unless the axial load is small or the
margin of safety is great. The approximate methods
will be sufficiently accurate for use in the preliminary
design of airplane members subjected to combined
loading, but all such members should be checked by: one
of the precise methods before being approved for use in
the final design.
PART IITHE DERIVATION OF THE PRECISE EQUATION
INTRODUCTION of the nature of these quantities. In the precise
formulas, as explained in the introduction to this report,
In Part I of this report various approximate formu these functions have no connection whatsoever with
las for the determination of stresses due to combined any angles, but are the limits of certain infinite series
bending and compression were studied and comparecl that can be most conveniently expressed as trlgonoto
each other and to th&precise formulas. None of the metric functions.
approximate formulas were found to be generally
applicable, and the use of precise formulas was recommended.
This part of the report gives the derivation
BASIC ASSUMPTIONS
of the precise formulas and the method of applying The basic assumptions from which the precise
them to practical design. formulas are developed are as follows:
The chief advantages of the precise formulas are as (1) Plane cross sections remain plane and normal to
follows: the longitudinal fibers after bending.
(1) The true bending moment at any section of the (2) The intensity of stress is proportional to the
beam can be obtained. strain throuBhout the member, and the ratios of stress
(2) The total deflection of any section of the beam to strain, the moduli of elaRticity, are the same in
can be obtained. tension and compression. •
(3) Very few assumptions are necessary. Those (3) Every longitudinal fiber is free to extend or
that are made are the ones generally made in develop contract under stress as if separate from the other
ing beam and column formulas. fibers.
(4) As shown in Parts III and IV of this report, the (4) The member is straight and homogeneous, and
deflections obtained from the formulas check experi the cross section of the member is uniform between
mental results in a very satisfactory manner, and much points of support.
better than the approximate formulas. (5) The axial load is applied iri such a 'way as to
(5) Once the size of the beam has b_een determined, develop no bending in the member due to eccentricities.
the compu~ations nece~sary with th~ precise formul~s I For a perfectly homogeneous material this requires
are less tedious than with the approximate formulas, if that the axial load be so applied as to pass through the
themargin of safety is not to be unnecessarily large. I centroid of each cross section of the undeflected mem
The chief disadvantages of the precise formulas are I ber.
as follows:
(1) The determination of the size members required
and the stresses involved are uependent on each other
in such a manner that the method of trfal and error
must be employed. This can, however, l:ie overcome
to a great extent by judicious approximations in the
preliminary design. Furthermore, it is felt that the
greater accuracy of the results is well worth any
increased labor of computation.
(2) Many engineers are unfamiliar with these
formulas. This is not considered sufficient reason for
neglecting them in.the face of their advantages.
(3) Speci;il complex functions and trigonometric
NOMENCLATURE
The nomenclature used is, with the exception of one
or two abbreviated. forms, standard in literature on
mechanics and is selfexplanatory with the use of the
figures. The abbreviated forms are described In the
appropriate places in the crerivation and should cause
no trouble.
The· conventions used for signs are those given on
page 3 of Part I of this report.
SCOPE OF DERIVATIONS
functions of numbers must be used. The complex The derivation will oe given in detail for the _most
functions needed and the trigonometric functions of common condition of loading encountered in airplane
numbers in the required range have been tabulated work, i. f' . , a member having a uniformly distributed
and are given in the appendixes to this report. These lateral l9ad in combination with an axial load causing
tables are arranged to obviate the necessity of trans cqmpression. The development of the' equations .for
forming numbers considered as angles measured in various other conditions of loading will be given in such
radians to angles measured in degrees, and vice versa. a way as to indicate the differences, so that they will
The tediousness of this operation was formerly a very serve as an aid to the designer should he . want · to
exasperating feature in the use of precise formulas. derive a formula for a loading condition which is not
(4) Previously the precise formulas were suspected included in this report. For brevity, computations
because of the presence in them of trigonometric involving only simple algebra or arithmetic are omitted
functions. This, however, was due to a misconception from all of the derivations.
(9)
10
CASE 1. AXIALLY LOADED STRUT WITH UNIFORMLY
DISTRIBUTED TRANSVERSE LOAD
Figure 3 shows a member supported at two points
and subjected to · a uniformly distributed lateral load,
an axial compression, and moments applied at . the
points of support.
p
FIG. 3
The expression for the moment at any point is
M=M +(M2Mi) _ wLx+wx
2
P 1 I L x 2 2 Y  
By making the usual assumptions of the beam
theory, l.to 3 above,
d2y
l1l= El dx2
whence Eld2y+P =M +(M2M1)· _ wLx+wx
2
dx2 y 1 L x 2 2
Differentiating twice with respect to x this becomes
d2 ( d2y) p d2y
dx2 EI dx2 + dxz =w
d2M P
or dx2+E1M=w
x
If wewrite,Jzfor J:i•i heing an abbreviation for..JEJ!
d2M 1
dx z +. J. ,.M= w
The solution of this differential equation is 1
M= C1 siny+C2 cos y+wj2 ___ _______ 2
Ci and C2 are constants of integration, sin!J; and cos
J arethe limits of infinite series in which the variable is
x
J
For purposes of computation they may be considered
as functions of tire angle~expressed in radians.
J
The series
. ~· _: _ (x/j)3 + (x/j)5 (x /j)7+
Sill j IS j LL I ~  I 7
and
cos ~ is 1 (x/j) 2 + (xfj)•  (i/j) 6 + .
J /~ I_!_ I~_
When x = 0, M = M1 and when x= L , M=M2, hence
Ci= Mzwj2_Mi  wj2 M2wj2 (M1wj2) cosL/j
. L L L
sui · ~ tan ~ sin ·~
J J J
C2= M1  wj2
For brevity, we shall write
D1=M1 wj2}
D2=M2 WJ2
The moment at any point is now
M (D2Di cos L fj) . x· x . :i  sinL/j s111T+D1 cos3+w3 ____ 3
To find the location of the section of maximum
moment, differentiate equation .2, equate the first
derivative to zero, and.solve
dM Ci x C2 . x
dx = 0= ycos37 sm3
tan~= 12!= D2D1 cos L/j 4 J C2 D _. L        
1 Slll T
The value of x determined from this equation must
lie between 0 am::l L. Otherwise, either M1 or M2
is the maximum on the strut.
The maximum moment may be found by substituting
the value from equation 4 in equation 3 and simplifyingo
Mmax.= Dix +wj2 ____ ____ ______ 5
coso
J
The deflection at any point is found by substituting
the value of M from equation 2 in equation 1 and
solving, whence
y=~(Mi + ( M2'i Mi)x  w~x +wt
L
DzDi cos~
~3~ sin~Di cos__:;_wj2) ____ 6
. L J J
sin~
J
The first derivative of equation 6 gives the slope of
the tangent to the elastic curve at any point,
. 1 (M2Mi) wL Ci x C2 . x)
i=p L 2+wx J cos3 +7 smy __ 7
If we have two continuous spans, as shown in Figure 4,
the slope of the tangent at the center support will be
the same ro·r lfoth spans, the member being continuous
over this support.
,.,,
P. ......
~ I f i i w+ ?1114 IHffl l . " .e,
·At R2, Xi = Li for the left, and x2== 0 for the righthand
span . Using subscripts to differentiate the symbols
I for t he respective spans and substituting the above
values in the expressions for slope at R,, we get
where
~ I
where
! See Hudson, The Engineers' Man_ual, par. 363, p. 58. 2 See Hudson, The Engineers' Manu"l, pA r. 111, p. 3li.
11
But at the center support i 1 = i 2• Substituting the
values for C1 , C'I, and C2 , equating 8 and 9, combining
terms, and simplifying, the following result will be
obtained:
[
LI LI J [L2 L
2 J M L , cosec ,  1 M L , cosec ...,..  1
_I_I JI !I +  3_ 2 )2 )2
I1 (];1)2 I2 (~ )'
I( ( 1 LI cot L i)) I( (1 L, cot L2))J
+ '\,:' ~ JI JI + M !:Y J2 J2 .
"
2
JI (~)' 2
J2 (~)'
M1LI °'I+ 2 M2{~ {3l + f2 (32} + M3L2 «2
I1 Ii I2 12
6E(yI y,) +6E(yay2)+W1L13 'Y1+W2L23'Y2 11 L1 L2 411 412 
If the deflected positions of the points of support
lie on ~traight line,
and the deflection terms drop out.
For airplane trusses y1, y,, and Ya may be computed
according to the usual methods for figuring truss deflection,
but with the assumption that the deflection
is due to the elongation of the wires alone, i. e., elongations
of the spars and struts may be neglected. The
[
LI LI J [ L2 L2 J tan   tan
= WILI3 2JI 2.iI + w2L23 2J2 2J2
lI C7iY 12 C~Y 
Multiplying this equation by 6, it becomes
MIL1aI+ 2M {!!J +f2 }+M3L2a2_wI£_1_3
lI 2 JI f3I l,!32 I2  411 'Yi
10 deflections are usually small and may generally be
omitted from the computations, though their effects
should always be considered, especially for spars continuous
over the center section where the deflected
positions of the points of support are obviously not· in
a straight line.
w2L2
3 + 41, 1'2     11
Where
Tables of sines, cosines, and tangents of L/J and of
a, (3 and 'Y in terms of L/J have been computed and
will be found in Appendix 3.
It often happens that moments are introduced at the
points of support of continuous members due to fittings
which are not concentric. The above formula may be
altered to provide for this condition, as follows :
In this equation M_2 and M+2 are the moments an
infinitesimal distance to the left and right of the point
of support, respectively. Equation lla contains an
extra unknown which necessitates another equation
for a solution. This is derived from the relation between
M_1 and M+2 , M+2 being equal to M_2 plus or
minus the eccentric moment M.. Care must be taken
with the sign of M.. It should be considered pm:itive
if it increases the moment from M_2 to M+2 as one goes
from left to right at the point of support.
When a truss deflects, the panel points do not necessarily
lie on a straight line. Equation 11 may be
modified to provide for differences in elevation of the
supports, as follows:
2222325f 3
CASE Ia.AXIALLY LOADED STRUT WITH NO
TRANSVERSE LOAD
The precise formulas above may be modified for
use with struts subjected to axial loads and end moments,
but no lateral load, by making w= O.
The position of the section of maximum moment
may then be found from
and
L
M 2  M 1 cos~
tan ~= J            12
J . L M 1 sin ~
J
M = ~    13 max. X
cos~
J _l_[ (Mz  MI) (M2lrf1 cos J) Yp M1+ L x . L
Sln ~
J
,;n }M, '°'} } 14
If one end is hinged, as often happens in the case of
struts used on airplane chasses, the moment at any
point is
Mz sin~ M= .!  15
Sill~
J
x being measured from the hinged end.
The location of the section of maximum moment is
x~i.!!2.     16
Mmax.= .M2£        17
8111 ~
J
v =~[Ml x _ ~~i~IJ ________ 1s
Sill · c
J
12
CASE Ib.PIN ENDED STRUT WITH AXIAL
LOAD AND UNIFORMLY DISTRIBUTED
TRANSVERSE LOAD
If both ends of a member subjected to an axial load
and a uniformly distributed transverse load ate hingec:l,
the section of maximum moment occurs at mid span.
Between the load and the right support we have
(
M2M1) W(La)
MR=M1 + L x L x
+ W(xa)Py ____________ 21b
As in Case I, take the second derivative of equation
21, whence
d2M Pd2y
Mmu. = wj2(1 ~)=wj dx2 =  dx2 2(1sec ~) 19
cos 2J or
Y
 !_ [wx2_ wLx+ wj2(1cos y) sin~
p 2 2 . L j
sin c
J
wi'( 1oo• J) ]'c 20
Johnson's approximate formula
Mo
M = 1PL2
l, lOEI
may be derived from formula 19 if the algebraic series
represented by the secant be substituted for the secant
term. The resulting form is
Mo
M= 5PL2
l 48EI
which reduces to Johnson's formula if 1/10 be substitu
·ted for 5/48.
d2M p
dx2 +EIM=O
The splution for this equation, when applied to the
segment to the left of the load is
ML=C1 sin j + C2cos } 22a
And for the segment to the right of the load
. x x
Mn= Ca sin T +c. cosT       22b
where j=JE_/;; and C1, C2, Ca, and C1 are constants of
integration.
From equations 21 and 22, we find that the deflection
is
(M,Mi) W(La) . x
PyL=M1+ L x L x  C1 sm7
x
C~cos 3 23a
CASE ILAXIALLY LOADED STRUT WITH CON and
CENTRATED TRANSVERSE LOAD
Figure 5 shows a span subjected to a concentrated
lateral load, an axial compression and ,moments applied
at each support. The method of derivation of · the
formulas for the moment in the span and for the threemoment
equation is analogous to that used in the preceding
case. The determination of . the constants of
integration is somewhat more difficult for the concentrated
than for the uniformly distributed load, but
even that is simple if the·.procedure outlined below is
followed.
FIG. 5
Since it Is impossible to write a single equation for
the moment at any point in the span for this type of
loading, one equation will be written for the segment
to the left of the 1011,d and another for the segmet'lt to
the right. The fact that two equations must be
handled instead of one appears, at first glance, to
render the solution much more difficult, but this will
not be found to be the case.
The expression for the moment at any point between
the left support and the load is
M M +(M2M1) W(La)x p
2
.
L= 1 L x L  Y   la
P M +(M2M1) W(La) +w' ( ) YR= 1 L x  L x , xa
 ca s.m 3x  c 4 cosx7  , · 23b
By differentiating equations 23, we find that the
slope is
c., cos x, +...c.,,. s.m x. __ 24a
J J J J
and
Pi&=M2;,M1 W(LL a) +w  ~·cos j
+ f sin j ~~ 24b
From the conditions of the structure, when x=O, !ff
=M1; whe1C~, YL= YR and iL = iR; and when x=L,
M=M2• These conditions are sufficient to determine
the four constants of integration, whfch will be found
to be
. L
M,M1 cos...,. a ( L a) C1 = J + Wj sin , cot ccot ~
. L 3 3 3
Sln,
J
a, cot:LJ
3
13
Differenti.ating equation 22 to find the section of found from the tables. It. will be noted that the leftmaximum
moment, hand side of the above threemoment equation is
So
Dividing equation 22 by cos~' we get
3
ML x =C1 tan ,.+C2
cos'.!;. ,
3
identical to that developed for the uniform load; such
differences as there are being on the righthand side
in connection with the terms which provide for the
load. This equation may therefore be treated in exactly
the same way as the threemoment equation in Case I
to allow for the effect of an eccentric moment at one
of the supports or for the deflection of the supports.
If a combined uniform and concentrated loading
were applied to a continuous strut the threemoment
equation could ·be arranged to provide for this condition
by including on the righthand side of the equation
terms sufficient to provide for each load, leaving the
lefthand side of the equ1J.tion unaltered.
or M
(c,2+ c22) x _ c2 'c 2 C 2 Formulas similar to those obtained in Case I may
mas. c, · cos3c, V 1 + 2  25a, be obtained for pinended struts, or for struts in which
one end is hinged and one restrained by substituting
Similarly, for the righthand segment, M1=M2=0 or M1=0 in the equations.
(
C 2+ C 2) x C 1 For a pinended column with a concentrated lateral
Mm••·= ~ cos 3=a!vC32+C42
 25b load in the middle of the span, the moment at any
It will be noted that for a single concentrated load
the section of maximum moment may come either to
the left or right of the load, depending on the position
of the load and the magnitude of the moments at the
supports. It may therefore be necessary to compute !
the values of tan ~ for both segments to ascertain in
3
which the section of maximum moment is located. If
.t is less than a, l!S found.for 'the lefthand segmf;mt, use
the form.ula for that segm~nt when computing the
maximum moment. If x is greater than a when
computed from the values of C1 and C2 the section of
maximum moment lies to the right o.f the load and its
location and magnitude should be computed from
Ca and C4. It is conceivable that the shape of the
moment curves to the left and right of the load will
be . such that the point of zero slope of each curve will
lie on the opposite side of the load from the curve itself.
Since the equation for the location of the section of
maximum moment is in reality simply a means of
determining 'the point when the slope of the moment
curve is zero, it will be found that for such a condition
the value of x determined from C1 and C2 will be greater
than a, while that determined from Ca and C4 will be
less. This indicates that the maximum moment will
be at the section where the load is applied an\f i;nay be
computed by substituting a for x; or that the concentrated
load 'does not cause a maximum in the span, in
whjch case either M1 or M2 ie the. maximum.
If the same procedure is followed for a continuous
span having concentrated side loads as was followed
in the case with the uniformly distributed side load,.
the following equation of three moments results:
M1L1a1+2M2.{L1.B1+ Lz.B.2}+ MaL2a2
I1 I1 I2 I2
[
. a, J [ . L2a2 J 6 w, · 2 sm ""7"' 6 ur . sin. L
=~ __ J_1 _ a1 +~ .J2 _ 2a2 26
11 . L, L1 I2 . L2 L2
sin ~ s1n "'7'
31 32
In this equation a and {J have the same values as in
the case of the uniformly distributed load and may be
point is
M=C1 sin]. 27
w·
· Ci== ~
2 cos 2J
where
The maximum moment is at mid span and is
W
.. L
 3 SIIl 2i
Mmu·= L
2 cos 2J
The deflection at any point is
y=~ 3 [w smI
.. x
2 cos 2J
The deflection at mid span is
Wj · L
2 tan 23 28
~1 · 29
y= :t [tan ~~J"·  30
This formula and an approximate formula derived
from it are used in Part III of this report to comp~te
the midspan deflection of the specimens which were
tested to sl).ow whether the theoretical formulas would
agree with practical results. For further discussion,
see Part III.
CASE 111.AXIALLY LOADED STRUT WITH
UNIFORMLY VARYING LATERAL LOAD
1.~t.
FIG. 6
Figure 6 shows a strut subjected to an axial compression,
a lateral load varying uniformly from zero
at one end to W at the other and to moments applied
at each support. The derivation of the formulas
for this condition of loading is practically identical
14:
in every step to the derivation for a uniform load. The threemoment equation is found in the same
The development will therefore be given in a briefer way as for the uniformly distributed load and may be
form. written
The moment at any point on the span is
  M1L1a1+ 2M2{[;_1f3,+~2f32}+M3L2a2= W1Ld1
2
[2({3, l)]
(
M=M1+ M2LM1) xW6L x + W6Lx3 Py ____ 31 I I I I I I I 2 2 I
The second derivative of this equation is
The solution for this equation is ·
Where i=JEJ, C1 and G2 are constants of integration.
When x = 0, M=M1 and whenx =L, M=!t!2, from
which it will be found that
. L
Sln .,.
J
C2=M1
Equation 32 does not offer a simple equation for
finding the location of the section of maximum moment,
as· its first derivative, when equated to zero, gives
x x Wj3
C2 sin yG1 cosy= L
which may be converted into the following expression
in ·terms of Ci, C2, and Wj3.
tan~=C1_ . (G,2+c22)~i3  33
J C2 G1C2Wj3±G22 .../L2 (C1~+c22)(Wj3) 2
It will be observed that two values of x will be found
from the equation, according to the sign used for the
radical term ·in the denominator. One value of x will
probably not lie cin the span, hence may be neglected.
The magnitude of the maximum moment i,f? found by
substituting the value of x obtained from equation 33
in equation 32 and solving. ·
The deflection at any point may be found from
1[ (M2M1) WLx Wx3
y=p Mi+  L x 6 + 6L
·'r x · •·W:xj2]
C1 sinyG1co1Jy L      34
And the slope at any point is
If the load varies from W at the left support to zeru
at the right, the equation is similar and may be written
CASE IV .GENERAL CASE
If, instead of a concentrated load W, a load of w dx
at a variable distance x from the origin had been used
in Case II, the following general form of the threemoment
equation would have been obtained:
In general, the material will be the same throughout
the length of the strut, so that E1 may be made equal to
E2, and the equation becomes
The threemoment equations developed in Cases I,
+f.
1
_2 sin 
1
::_ WLj2J  · 35 II, and III may be obtained from equation 38 or 38a
by integrating and simplifying the resultant expressions.
15
CASE V.STRU'J,' SJJJ.UECTED TO UNIFORMLY Where
DISTRIBUTED J.ATERAL LOAD AND AN AXIAL
TENSI.oN
Di= M1 + wj 2
D2.= M2 + wj2
The foregoing formulai:; have all been derived for
cases where the' axial l~i;Ld c~used compression in the
member. This is the more important case, from a
structural standpoint, since .an axial compression
increases the bending on the strut an'd accelerates i~s
failure; whereas an axial tension tends to straighten
the member out and retard its failure .
. For this reason it is more conservative in the latter
case to compute the stresses due to the . axial and transverse
loads independently and to add· the results.
The formulas will be given for the case of a uniformly
distributed transverse load in combination with
an axial tension, so that a designer may have a basis for
deriving the equations for otP,er loading conditions if
he should desire to do so.
The same procedure may be used as in the case of
axial compression, except that  P should be substituted
for P. The equation for the moment at any point on
the span, instead of equation I, .pa11:e 10, becomes
M=Mi+(μ2LM1
) x~Lx +w;
2
+Py
and the second derivative is
d2M P   M=w
dx2 EI
The solution of this differential equation is
So
M = 61 sinh f+C2 cosh }wj2, where j=.JEJ
. L
sinh ~
J
C2= M1+wj2
It will be noted that the circular fo.nctions occurring
in the former cases have been replaced by hyperbolic
functions and that some of the signs have been changed.
These changes are brought about by the ·change in
sign of the second term in the differential equation and·
will require careful attention when deriving formulas
for other types of loading.
The expression for the maximum moment is similar
to that for axial compression. i
The location of the section is at x, where
L
tanh .:!;.=Di cosh y D2
J D1 sinh J
and
D1 :2 M max. =WJ
cosh x
J
The deflection at any_point is
Y=p1 _( c is.m hyx +C2 cosb xy wJ. 2  M 1  ·cM2L M1) x
+wLx_wx2
)
2 2
The slope at any point is'
._1 (C1 hx +c2 . • h x i  p ~ cos ~ · ~ sm ~
J J J J
The threemoment equation is similar to that for
axial compression, when a, {J, and 'Y are replaced by
ai., fh, and 'Yb, where
f3b
'Yb=
MiL1ab1+ 2M {bf3 + ~{3 }+M3L2ab2
11 . 2 I I hi I 2 b2 12
wj £13 W2 L23
= 4h'Yb1+ 412 'Yb2
Values of ai., f3h, and 'Yb are _tabulated in the
Appendix of this report.
Values of the hyperbolic sines, cosines, or tangents
may be found in ·almost any fairly complete set of
mathematical tables.
Terms may be added to provide for the deflection of
the supports in this case just as well as in the case of
axial compression.
For other conditions of loading the development of
the equations will. be left to the designer, with the
warning that particular attention must be paid to
signs, especially wben obtaining a solution for the
differential equation.
Should the case arise where a spar is subjected to
compression in one span and tension in the next, the
precise threemoment equation may be altered to
provide for it'by using the coefficients based on circular
functions with the terms relating to the span under
compression and the coeffieients based o_n the hyperbolic
functions for those relating to the span in tension.
The moment!! and stresses within the spans are then
found by use of the formulas for axial compression or
tension as the case may be. This condition seldom
occurs in practice, a\though it may be ·found in an
airplane wing spar due to the action of the drag
truss stresses, which may oppose and overcome those
from the lift truss in one rJii:y but not in the next.
16
CRITICAL POINTS IN THE PRECISE
FORMULAS
FORMULA FOR MOMENT IN ANY POINT IN
THE SPAN
Each of the precise formulas is difficult to solve for
certain values of L /j. In some cases the formula will
take an indeterminate form such as ~or g ;. in others the
functions vary so rapidly· that reliable results can not
be obtained with ordinary straightline interpolation.
For instance, if we in;yestigate the general expression
for the bending moment at any point in a span under
combined bending and compression, i.e.,
(
D2D1 cos!;) . x x .
M = L J sm...,. + D1 cos...,. + WJ2
sin ...,. J · J
J
we immediately see · that the term in parentheses
becomes infinite when sin L/j is zero. Since sin L /j
= 0 when L/j = ·o, .,,., z.,,., etc., it is apparent that these
points are critical ones when applying the formulas.
It is possible., however, to compute the moment for
values of L /j near the critical point, plot the results,
and obtain a value of the moment at the dritical point
from t he curve drawn through the points. Fig·ure
7 gives t he results of solving the equation for several
values of L /j near v and shows that the bending moment
when L /j is .,,. is about 42,100 in. lbs. in this
case.
.. = ~AM .. "' ,_ .  . "
I ..
r
i ....
·
·i. 2 ~ .,.
w. = '1 . . ,,..,, .. ..u.
/ / . ,,., '"""' '. .,,,,, ~ 
'
... •
.. I
I
t I
v
:1 1;
/
q /
.. _17
/  '
i·   cu,.v.e .s:.102 jl(rrktbO/l o/  /1'117%//11//l'/1 '/11eAt /n ¥'°"
#?~ writrlion m '{r
/.0 / .S ~v .I .J .cl" I 1>w I I 1..:1.5
. ~, ,.,,, ~ l'/ ..,.d"l I I I I I
FIG. 7
FORMULA FOR MAXIMUM MOMENT IN A SPAN
If we consider the formula for obtaining the maximum
moment in a span, we see that it becomes indeterminate
when D1 = 0. That is,
D,
Mmax.= X +wj2
cos ,
J
15 ecomes Mmax. = 0O + WJ.2 when D 1= 0, since tan Jx as
found from
L
D2  D1 cos ...,. _ ____ J,,_ is oo
D
. L
I Sill ...,.
J
x .,,. ·x ·
y =z and cos y=O. It is possible to dodge this complication
by observing that}=~ and using the general
expression for moment at any point in the span, which
becomes
M D, ·2
= 'L + WJ
Rill ...,.
J
For points near D1= 0 the tangent is varying very
rapidly and it is impossible to obtain t he values of ~
J
and cosy with sufficient precision by ordinary methods
of interpolation.
The simplest method of attack in such a case is to
interchange D, and D2 in the formulas for M ma•. In
this case x will be measured from t he right hand end
of the span; that is, the interchange of D1 and D2 is
equivalent to turning t he beam end for end. Unkss
L /j is near .,,. iu value, this dodge will alter the value of
x/j sufficientJy to permit a solution without recourse
to any special formulas.
A second method depends on the fact that
1 ~·   A=sec A =·v'tan2 A+ l
cos
Applying this r elationship to the expression for t.he
maximum moment in t he span, we have
Substituting for tan :::, ' its value,
J .
L
D2D1 cos ""7
J
D . L
l Sill T
and simplifying, the expression becomes
Mmu. = ~ VD12+ D22ZD1 D2 cosy +wP
SIIl ...,.
J
which does not involve the quantity x/j. · By using
this formula it is possible to find the maximum moment
without solving for x/j, but the location of this moment
is not found, and it is sometimes difficult to tell which
17
of the two square roots of the expression under the
radical sign should be used.
A third method, which is approximate but sufficiently
accurate when x/j is nearly equal to 7r/2, depends on
neglecting the 1 under the radical in the · expression
It can also be seen from the symmetry of the loading
that P1 = P2, L 1 =L2, etc., so that in this case the
moment at the center support will become infinite when
_ 1_=.,Jtan2 A+1 and assuming ~A = tan A. · This
cos A cos . or
whence
and
1L/j cot_!d_j=l
L /j cot L/.i1
1L/j cot L /j = L /j cot L/j1
L/j cot L /j = 1
L/j = tan L /j
is merely another way of stating the assumption that
cos A = cot A, which is as valid for values of A dose
to 7r/2 as the similar assumption that sin A = tan A for
values, of A close to zero. If the above assumption is
made, the formula for the maximum moment in the
span becomes This condition is fulfilled when L /j is approximately
1lf = D1 tau :'._ + wJ·2 · max. J
Substituting the expression for tan x/j iii this l'~l"°"l!!!]~~~f!:J:ttttttfi~~~~tjtJjJ:J formula it becomes I /1 = IK'il'fA  fl'!! '.I
.l'lfmax. =
D,  D1. cos .L..,..
~~.1 +wj2
. L
Sill c
J
The first two methods are simply algebraic
transformations of the equations and
give correct results for all values of x/j .
The third method involves an error of less
than 1 per cent in the first term of the
expresison for M max. when x/.i is between 1.45
and 1.70.
THREEMOMENT EQUATION
 ,,_
'

~ 1
'"b
'·~ ., I
I
I
J
I
I
I
I/
i J
The threemoment equation also has critical
points and for certain values of L /j appears
to give infinite results.
''fl9!.+++++++++tt,!,1~~t1f1if/'++++t+1
If we write the threemoment equation in a
form that is readily solvable for M2, i. e.,
W1L13'Y1+ W2Li3'Y2_ M1L1a1_ I'lf3L2a2
M,= 411 4/2 / 1 / 2
2 {1:..1
fJ1 + 0. fJ2 } 11 12
We find that M, appears to become infinite
when L/j = 71', since ~. {J, and 'Y are each
iμfinite. As a matter of fact this expression
really becomes .::' and mav be evaluated in
00 . •
 0 ~/J Afl .ll5"
FIG. 8
the ordinary manner for solving indeterminate forms 4.49, as t itn 4.49=4.45. The moment at the center
by differentiation. support will n'Ot be infinite when L/j = O, although tan
It is apparent from the above equation for M2 that 0=0, as 6 cot O is not equal to 1.
the righthand side becomes infinite when the denomi When the beam is continuous over more than two
nator is zero. If we investigate this condition, substi spans, the problem of solving for the moments at the
tuting the trigonometric functions for fJ, we find that the> support when L /j iμ pn£l. ~p an is equal to 71' b~_<;p_mes
clenominator becomes zero when · more involved and has not been attempted by .. the
L L writers of this report.
1 ~ cot 1. p L The critical points encountered ill the use of the }I · . Jt I I
L L P L precise formulas for a member under combilled bending
....) cot ~ 1 2 2
3
J and compression are apparently critical only in that
2 2 I they require . specjal·.consideration when the fOFmulas
Figure 8 is a curve showing the variation of moment are being solved. Similar points can be found for the
at the intermediate point of support for different other loading conditions, bf t !N1ey wi]l not be enumervalues
of L/j, for the span and loading shown on the I ated here. The kn_ owledge of their existence should
_figure. It is obvious from this figure that the curve be sufficient to put the designer on his guard when
of M2 is continuous through L/j=1f'. applying the formulas at points where the trigonometric
18
functions are changing rapidly or at points where any
of the terms in the formulas approach zero or infinity.
No difficulty will be encountered when applying these
formul.as so long as L/j in any span is not equal to or
greater than 1r. When L /j is greater than .,,. the valtie
of the maxiumm moment in any span if found from
the value of x/j should be cpecked by the other expression
for the maximum moment, whii:h is based on
L /j only. There are at the present time no known
test data to indicate whether or not these formulas
are in reasonable agreement with a\:tual results for
loads and spans having L /j greater than .... It is
therefore recommended that spar sizes to be used in
airplane wings be so designed that L /j in every span
will be less than .,,. .
METHOD OF PROVIDING FOR VARIATION IN
THE MOMENT OF INERTIA
In an airplane wing cellule the drag truss bays are
generally shorter than those in the lift truss, there
usually being two or three drag bays to one lift truss
bay. The stresses in the drag or antidrag wires
produce axial loads in the spars, so that the value of P
ean not be taken as constant thro1ighout a lift truss
bay. · In addition to this variation in the axial load
the spar sections themselves are often varied in the
different ·drag bays, so that the moment of inertia is
not constant. The precise formulas developed in the
foregoing pages are therefore not accurate for these
conditions, as they assume a . conistant axial load and
constant moment of inertia between supports. The
following method might be used for approximating
the moments on the spars under such circumstances.
.L,
FIG . 9
J."'
P'" I"'
·I
Figure 9 shows one bay of a lift truss of length L1,
which is subdivided by the drag truss into three bays
of length L', L", and L'". The axial loads are P' ,
P" , and P"' and the cross section will be assumed to
change at each drag truss panel poh1t so that the
moments of Inertia are I', I", and ! 111
•
Let
and
P' L'·+ P" L" + P" 'L"'
L' +L" +L'"
I' L' + !" L" + I"' L'"
L' + L" + L'"
and compute j1 in the .usual way. This value of j
should be used in the threemoment equation when
computing Mi, M2, etc. With M 1 and M 2 known,
compute the moments at the drag truss panel points,
i. e., M', M", for the above value of j. Using these
values of M 1, M', M", and M2 as the end moments,
M1 and M2, consider each drag bay as a separate span
and apply formulas 4 and s;usingthe value of j obtained
from the P and I for the bay being considered, not
the weighted values used above. The quantities
obtained in this way, while admittedly approximations,
wj)l be somewhat more conservative than those
computed for the average load and average moment
of inertia.
NOTES ON THE USE OF THE PRECISE
FORMULAS
It is recommended that at least four significant
figures be used in all computations involving the
precise formulas. In preliminary investigations or
for the purpose of obtaining a rough check on a spar,
three figures will be sufficient but, since the final result
of several of these formulas depends on small differences
between large quantities, three significant
figures will often give misleading results. This fact
should be borne in mind and the number of significant
figures necessary to give the required precision in the
results should be used. This matter is especially
important when the value of L /j is near.,,., as the functions
a, (3, and 'Y are all changing rapidly in that range.
Special care must be taken to use the proper signs
throughout the computations or ~eri ons errors will
result. This is particularly true fo the case of the signs
of the terms for loads and deflections. The con ventions
for signs are given on page 3 of Part I of this
report.
In applying the precise formulas to design, it is
necessary to decide upon a size of member before the
final values of the bending moments can be obtained.
This makes the process of design a matter of successive
trials, but by first computing t he bending moments
and axial loads without allowing for secondary stresses
and using those moments and loads suitably modified
by the judgment of the designer on the first trial, the
number of trials needed to obtain a satisfactory design
should not be excessive.
In discussions of the precise methods of computing
stresses due to combined loadings in this report and
by other authors (as Cowley and Levy in Aeronautics
in• Theory and Experiment, etc.), failure is usually
assumed to mean failu!'ll due to elastic instability or
"buckling." Usually, before such failure would
occur in practice, the. member would have failed by
rupture of the material due to excessive unit stresset>,
and statements regarding the criteria for failure must
be read with these facts in mind. The criteria fur
failure in buckling implicitly assume that the material
has a constant finite modulus of elasticity and an infinite
proportional limit and ultimate allowable stress.
Of course, no engineering material has such properties,
but the precise formulas · and resulting criteria for
buckling failU're are nevertheless very useful in determining
the loads under which failure by rupture of the
material is \.4.kely to occur.
APPENDIX I
Table A gives values of sines, cosines, and tangents
of angles in terms of radians between 0.00 and 3.50
radians. This table may be used in the same way
as any table of trigonometric functions, the argument
being in terms of .radians instead of degrees ti.nd minutes.
Attention is called to' the fact that the signs
of the functions are given in the tables, and care should
be ·taken to employ the correct sign when using the
val~es in the precise equations.
Table B gives values of a, {3, and 'Y in terms of radians
between 0 and 27r. It should be noted that the
increment to the argument is not constant but th.at it
val'ies in different parts of the table, .being least where
the rate of chan.ge of a, {3, and 'Y is greatest, so that
care is required when an interpolation is made.
The functions given in Tables C are the values of
a, {3, and 'Y to be used when the axial load causes
tension in the spar. They are based on the hyperbolic
functions and are differentiated from the values for
use with axial compressive loads by the subscript h,
being written ab, f3b, and 'Yb ·
Tables of hyperbolic series, cosines, and tangents
are not included in this set, as .they may be found· in
numerous collections of mathematical tables and, as
th!=J argument is practically always expressed in radians,
these tables are ·i'mmediately useful.
TABLE A.Natuml sines, cosines, and tangents. of
angles in radians
· x· I Xin
.m decimals
radians in degrees
Si.ne X Cosine X Tangent X
0.00 0. 000 o. 00000 I. 00000 0. 00000
0. 01 o. 573 0. 01000 0. 99995 o. 01000
0.02 ]. 146 o. 02000 0. 99980 0. 02000
0.03 I. 719 0, 03000 0. 99955 0. 03000
0.04 2. 292 0. 03999 o. 99920 0. 04002
0. 05 2. 865 0. 04998 o. 99875 0. 05004
0.00 3.438 o. 05996 o. 99820 0. 00007
0. 07 4. 011 o. 06994 0. 99755 o. 07012
0. 08 4. 584 o. 07991 0. 99680 o. 08017
0.09 5. 157 0. 08988 o. 99595 o. 09024
0.10 5. 730 0. 09983 o. 99500 0. 10034
0.11 6 .. 303 0. 10978 o. 99396 0. 11045
0. 12 6. 875 0. 11971 o. 99281 0. 12057
0.13 7. 448 0. 12963 o. 99156 0. 13073
0. 14 8. 021 .0. 13954 o. 99022 o. 14092
0.15 8. 594 0. 14944 o. 98877 0. 15114 .
0.16 9.167 o. 15932 o. 98723 0.16138
0. 17 9. 740 o. 16918 0. 98558 0.17165
0.18 10. 313 o. 17903 0. 98384 o. 18197
0. 19 10. 886 0.18886 o. 98200 o. 19232
0.20 II. 459 0.19867 o. 98007 0. 20271
0. 21 12. 032 0. 20846 0. 97803 o. 21314
0. 22 12.605 o. 21823 o. 97590 ·O. 22362
0. 23 13: 118 o. 22798 0. 97367 o. 23414
0. 24 13. 751 o. 23770 0. 97134 o. 24472
0. 25 14. 324 o. 24740 0. 96891 0. 25534
0. 26 14. 897 o. 25708 0. 96639 o. 26602
0. 27 15. 470 0. 26673 0. 96377 0. 27676
0.28 16. 043 0. 27636 0. 95106 0. 28756
·o.29 16. 616 0. 28595 o. 95824 0. 29841
TABLE A.Natural · sines, co:iineli, and tangents of
angles in radiansContinued
raxd iLsnn sI .dexm mina ls Sine X Cosine X · Tangent X
m dt!grees
0.30 17. 189 . 0. 29552 0. 95534 0. 30934
0. 31 17. 762 0. 30606 0.95233 0. 32032
0.32 18 .. 3~5 0.3H57 0. 94924 0. 33139
0. 33 18. 907 o. 32404 o. 94604 0. 34253
0. 34 19. 481 0. 3:1349 0. 94275 0. 35374
0.35 20;054 0. 34i90 0. 93937 0. 36503
0.36 20. 626 0. 35227 0. 93590 0. 37640
0. 31' 21.199 o. 36162 0:93233 o. 38786
0. 38 21; 772 o. 37092 0. 92866 0. 39941
0. 39 2.2. 345 O; 38019 o. 92491 0. 41105
0." 40 22. 918 0. 38942 0. 92100 0. 42279
0. 41 23. 491 0. 39)!61 0. 91712 0. 43463
0. 42 24. 064 0. 40776 o. 91309 o. 44657
0. 43 24, 637 0. 41687 0. 90897 o. 45862
0. 44 25. 210 0. 42594 o. 90475 o. 4.7078
0.45 25. 783 o. 43497 I 0. 90045 0.48300
0.46 26. 356 o. 44395 0. 89605 .0. 49545
0. 47 26. 929 0. 45289 0. 89157 0. 50796
0. 48 27. 502 0.46178 0. 88699 0. 52061
0. 49 28. 075 0. 47063 o. 88233 o. 53339
0. 50 28. 648 0. 47943 0. 87758 o. 54630
0. 51 29: 221 o. 48818 0. 87274 o. 55936
0.52 29. 794 0. 49688 0.86782 0. 57256
o. 53 30. 367 0. 50553 o. 86281 0. 58591
0.54 30. 940 0. 51414 0. 85771 o. 59943
0. 55 31. 513 0. 52269 o. 85252 o. 61310
0.56 32. 086 0. 53119 o. 84726 o. 62695
0.57 32. 658 0. 53963 0. 84190 o. 64097
0.58 33. 232 0. 54802 0. 83646 o. 65517
0.59 33. 80,5 0. 55636 0. 83094 o. 66955
0.60 34. 377 0. 56464 0. 82534 0. 68414
0. 61 34. 950 0. 57287 0. 81965 o. 69892
0.62 35. 523 0. 58104 0. 81388 0. 71391
0. 63 36. 096 0. 58914 0. 80803 0. 72911
0.64 36. 669 0. 59720 0. 802!0 o. 74454
0. 65 37. 242 0. 60519 0. 79608 o. 76021
0.66 37. 815 0. 61312 0. 78999 o. 77611
0. 67 38: 388 0. 62099 0. 78382 0. 79226
0. 68 38. 961 o. 62879 0. 77757 0.80866
0.69 39. 534 0. 63654 0. 77125 o. 82533
0. 70 40. 107 0. 64422 0. 76484 0. 84229
0. 71 40. 680 o. 65183 0. 75836 o. 85953.
0. 72 41. 253 o. 65938 0. 75181 0. 87707
0. 73 41; 825 0. 66687 0. 74517 0. 89492
I 0. 74 42. 399 0. 67429 0. 73847 o. 91309
0. 75 42. 972 o. 68164 0. 73169 o. 93160
0. 76 43. 545 0. 68892 0. 72484 o. 95045
0. 77 44. 118 0. 60014 0. 71791 0. 96967 o. 78 44. 691 0. 7032& . o. 71091 0. 98926 I
0. 79 45, 264 0. 71035 0. 70385 1. ()0924
0. 80 45. 837 o. 71736 o. 69671 I. 02964
0.81 46. 410 o. 72429 0.68950 ]. 05046
0.82 46. 983 o. 73115 0. 68222 I. 07171
0."83 47. 556 o. 73793 0. 67488 ]. 09343
0. 84 48. 128 o. 74464 0. 66746 1.11563
0.85 48. 701 o. 75128 0. 65998 1.13834
0. 86 49. 274 0. 75784 0. 65244 1.16155
0.87 49. 847 0. 76433 0. 64483 1.18533
0. 88 50. 42() 0. 77074 o. 63715 I. 20967
0. 89 50. 993 0. 77707 o. 62941 I. 23460
0.90 51.566 0. 78333 0. 62161 I. 26016
0. 91 52.139 0. 78950 0. 61375 1. 28637
0. 92 52. 712 0. 79560 0.60582 1.31326
0. 93 53. 285 Q. 80162 0.59783 I. 34088 !
0.94 53. 858 0. 80756 0. 58979 1.36923
o. 95 54. 431 0. 81342 0. 58168 I. 39838
0.96 55. 004 0. 81919 . 0. 57352 I. 42836
0. 97 55. 577
I
0:82489 0. 56530 I. 45920
0.98 56. 150 0.83050 0. 55702 I. 49096
0.99 56. 723 0. 83603 O; 54869 ]. 52368
(19)
20
TABLE A.Natural sines, cosines, and tangents of TABLE A.Natural sines, cosines, and tangents of
angles in radiansContinued angles in radians Continued
Xin Xin
radians decimals Sine X Cosine X Tangent X
in degrees
xm I Xin. I I
radians .decimals SineX I Cosine X TangentX
m degrees
1.00 57. 296 0.84147 o. 54030 I. 55741 1.80 103. 132 0. 97385 0. 22721 4. 28627
1.01 .57.869 0.84683 o. 53186 1. 59221 h81 103. 705 0. 97152 0. 23693 4.10050
1.02 58. 442 0.%211 o. 52337 1. 62813 1. 82 104. 278 0. 96911 0. 24664 3. 92937
1.03 59. 015 0. 85730 0. 51482 1. 66525 1.83 104. 851 0. 96659 0. 25631 3. 77118
1.04 59. 588 0. 86240 0. 50622 1. 70361 1.84 105. 424 0. 96398 0. 26597 3. 62450
1.05 60.161 0.86742 o. 49757 1. 74332 1.85 105. 997 0. 96127 0. 27559 3.48806
1.06 60. 733 o. 87236 0.48887 1. 78442 1.86 106. 570 0. 95847 0. 28519 3.36083
1. 07 61. 306 0.87720 0. 48012 1. 82703 1. 87 107; 143· o. 95557 0. 29476 3. 24188
1.08 61. 879 0. 88196 0. 47133 1. 87122
1.09 62. 452 0. 88663 0. 46249 1. 91710
1.88 107. 716 0. 95257 0. &,0430 3.13039
1.89 108. 289 0. 94949 0.31381 3. 02566
1.10 63. 025 0. 89121 o. 45360 1. 96476 1.90 108. 862 0. 94630 0. 32329 2. 92710
1.11 63. 598 0.89570 o. 44466 2. 01434 1. 91 109. 435 0. 94302 0. 33274 2.83414
1.12 64.171 0. 90010 o. 43568 2. 06595 1.92 110. 008 0. 93964 0. 34215 2. 74630
1.13 64. 744 0. 90441 0. 42666 2.11975 1. 93 110. 581 0. 93618 0. 35153 2. 66.116
1.14 65. 317 0. 90863 o. 41759 2. 17588
1.15 65.890 0. 91276 o. 40849 2. 23449
1.94 111.154 0. 93262 0. 36087 2. 58433
1. 95 111. 727 0. 92896 0. 37018 2. 50947
1. 16 66. 463 0. 91680 0. 39934 2. 29580 1.96 112. 300 0. 92521 0. 37946 2. 43828
1.17 67. 036 0. 92075 0. 39015 2.35998 1.97 112. 873 o. 92137 0. 38869 2.37049
1.18 67. 609 0. 92461 0. 38092 2.42726 1.98 113. 446 0. 91744 0.39788 2. 30582
1.19 68.182 0. 92837 0. 37166 2: 49790 1.99' 114:·0111 0. 91339 0. 40703 2. 24408
1. 20 68. 755 0. 93204 0. 36236 2. 57215 2. 00 114. 592 0. 90930 0.41615 2.18504
1. 21 69. 328 0. 93562 o. 35302 2. 65033
1.22 69. 901 0. 93910 0. 34365 2. 73276
2. 01 115. 165 0. 90509 0. 42522 2.12853
2. 02 115. 738 0. 90079 0. 43425 2. 07437
1.23 70. 474 0. 94249 o. 33424 2. 81982 2. 03 116. 310 0. 89641 0.44323 2.02242
1. 24 71. 047 0. 94578 o. 32480 2. 91194 2.04 116. 883 0. 89193 0.45218 1. 97252
1.25 71.620 0. 94898 0. 31532 3. 00957 2.05 117. 456 0. 88736  0. 46107 1. 92456
1.26 72. 193 o. 95209 0. 30582 3.11328 2.06 118: 029 0. 88270 0.46993 1. 87841
1. 27 72. 766 o. 95510 o. 29628 3. 22363 2. 07 118. 602 0:87797 0. 47873 1. 83396
1.28 73. 339 0. 95802 0. 28672 3. 34135 2. 08 119.175 0. 87313 0.48748 1. 79112
1.29 73. 912 0. 96084 0. 27712 3. 46721 2.09 119. 748 0. 86822 0. 49619 1. 74977
1.30 74. 485 0. 96356 0. 26750 3. 60210
1. 31 75. 057 0. 96618 0. 25785 3. 74708
2.10 l~.321 0. 86319 0. 50485 1. 70984
2.11 1 . 894 0. 85812  0. 51345 1. 67127
1.32 75. 630 0. 96872 o. 24818 3. 90335 2.12 121. 467 0.85294 0. 52200 1. 63395
1.33 76. 203 0. 97115 0. 23848 4. 07231 2.13 122. 040 0. 84768 0. 53051 1. 59785
1.34 76. 776 0. 97348 0. 22875 4. 25562 2.14 122. 613 0. 84233 0. 5.1896 1. 56287
1. 35 77. 349 0;97572 0. 21901 4. 45523 2. 15 123. 186 0. 83690 0. 54736 1. 52898
1.36 77. 922 0. 97786 o. 20924 4. 67344 2.16 123. 759 0. 83138 0. 55569 1. 49610
1.37 78. 495 0.97~1 0.19945 4. 91306
1.38 79.068 o. 98185 0.18964 5.17744
2.17 124. 332 0. 82579 0. 56399 1.46419
2.18 124. 905 0. 82010 0. 57222 1. 43321
1.39 79 . .641 o. 98370 0.17981 5.47069 2.19 125. 478 0. 81434 0. 58039 1.40310
1.40 80.214 o. 98545 0.16997 5. 79788 2.20 126. 051 0. 80849 0. 58850 1.37382
1. 41 80. 787 o. 98710 0. 16010 6.16537 2. 21 126. 624 0. 80258 0. 59656 1.34534
1.42 81.360 0. 98865 0.15023 6. 58110 2. 22 127. 197 0. 79657 0. 60455 1. 31761
1. 43 81.933 0. W<llO 0.14033 7. 05546 2. 23 127. 770 0. 79048 0. 61249 1. 29060
1.44 82. 506 0. 99146 0.13042 7. 60182
1.045 83. 079 0. 99271 0.12050 8. 23810
2. 24 128. 343 o. 78432· 0. 62036 1. 26429
2.26 128. 916 0. 77807 . 0. 62818 1. 23863
1.46 83.M2 0. 99387 0.11057 8.98862 2. 26 129. 489 0. 77175 0. 63593 1. 21360
1.47 84. 225 0. 99492 0.10063 9. 88740
1.48 .84.798 . 0. 99588 0.09067 10. 98338
1.49 85. 371 0. 99674 0.08071 12. 34991
2. 27 130, 061 0. 76536 0. 64361 1.18916
2.28 130. 634 0. 75888 0. 65124 1.16531
2. 29 131. 207 0. 75232 0. 65879 i.14lli9
1.50 85. 944 0. 99749 0. 07074 14.10142
1. 51 86. 517 0. 99815 0.06076 . 16. 42811
1. 52 87. 090 0. 99871 0.05077 19. 6696
1. 53 87. 663 0. 99917 0.04079 24.4986
1.54 88. 236 0. 99953 0.03079 32. 4513
1. 55 88.808 0. 99978 0.02079 48.0803
1. 56 89. 381 0. 99994 +0.01080 92. 6238
1.57 89. 954 1.00000 +o.oooso 1, 275. 04
1.58 90. 527 0. 99996 0.00920 108.661
2. 30 131. 780 0. 74571 0. 66628 1.11921 I
2. 31 132. 353 0. 73902 . o. 67370 1.09694
I
2. 32 132. 926 0. 73224 0. 68106 1.07514
2.33 133. 499 0. 72538 0. 68834 1.05381
2.34 134. 072 0. 71847 0. 69556 1.03292
2. 35 134. 645 0. 71147 0. 70271 ~L 01247 I 2. 36 135. 218 o .. 70441 0. 70979 0. 99242
2.37 135. 791 0. 69728 0. 71680 0. 97276
2.38 136. 364 0. 69007 0. 72374 0. 95349
1. 59 91.100 0. 99982 0.01920 52.0676 2. 39 136. 937 o. 68281 0. 73060 0. 93457
1.60 91. 673 0.99957 0. 02920 34. 2329 2.40 137. 510 0. 67547 0. 73739 0. 91602
1. 61 92. 246 0. 99923 .0.03920 25.4950 2. 41 138. 083 0. 66806 0. 74411 0. 89779
1. 62 92. 819 0. 99879 0. 04919  20.3073 2. 42 138. 656 0. 66058 0. 75076 0. 87989
1. 63 93. 392 0. 99825 0.05917 16.8712 2.43 139. 229 0. 65304 0. 75733 0. 86230
1.64 93. 965 0. 99760 0. 06915 14. 4270 2.44 139. 802 0. 64544 0. 76383 0. 84502
1. 65 94. 538 0. 99687 0.07912 12. 59926 2. 45 140. 375 0. 63777 0. 77023 0. 82801
1.66 95. 111 0. 99602  0.08908 11.18059 2.46 140. 948 0. 63003 .,0. 77657 0. 81130
1. 67 95. 684 0. 99508 'O. 09904 ~10.04724
1.68 . 96. 257
I
0. 99404 0.10899  9.12076
1.69 96.830 0. 99290 0.11892 8.34925
1. 7() 97.403 0. 99167 ...:.o. 1288.5 7. 69660
1.11 97. 976 0. 99033 0.13876 713723
2. 47 141. 521 0. 62224 0. 78283 0. 79485
2.48 142. 094 0. 61438 0. 78901 I 0. 77866
2. 49 142. 667 o. 60646 0. 79512 0. 76272
2. 50 143. 240 0. 59847 0. 80114 0. 74703
2. 51 143. 812 0. 59043 0. 80709 0. 73155
1. 72 98. 549 0. 98889 0.14865 6.65245 2. 52 144. 385 0. 58233 0. 81295 0. 71632
1. 73 99. 122 0. 98736 .o. 158.54 6. 22809 2. 53 144. 958 0. 57417 0.81874 0. 70129
1. 74 99. 695 0. 98572 0.16840  5. 85353 2. 54 145. 531 0. 56596 0. 82444 0. 68647
1. 75 100. 268 0. 98399 0.17825 5. 52037 2. 55 146. 104 0. 55769  0. 83005 0. 67186
1. 76 100. 841 0. 98215 0.18808 5. 22209 2. 56 146. 677 0. 54936 0. 83559 . 0. 65744
1. 77 101. 414 0. 9!!023 0.19789 4. 95340 2.57 147. 250 0. 54097 0. 84104  0. 64322
1. 78 101. 987 0. 97819 0. 20768 4. 71010 2. 58 · 147. 823 . 0. 53253 0. 84641 0. 62917
1. 79 102. 559 0. 97607 'O. 21746 4.4S866 2. 59 148. 396 0. 52405 0. 85169 0. 61530
I I
'\
21
.TABLE A.Natural sines, cosines, and tangents of TABLE A .Natural sines, cosines, and tangents of
angles in radiansContinued angles in radiansContinued
Xin Xin
radians decimals SineX Ciisine X in degrees
2. 60 148. 969 0. 51550 0. 85689
2. 61 149. 542 o. 50691 0. 86200
2. 62 150. 115 0. 49827 0. 86703
2.63 150. 688 0. 48957 0.87198
2.64 151. 261 0. 48082 . 0. 87682
2. 65 151. 834 o. 47204 0. 88158
2. 66 152. 407 0 .. 46319 0. 88626
2. 67 152. 980 0. 45431 0.89084
2.68 153. 553 o. 44538 0.89534
2. 69 154. 126 0. 43640 0. 89975
2. 70 154. 699 o. 42738 0. 90407
2. 71
I
155. 272 o. 41831 0. 90830
2. 72 155. 845 0. 40922 0. 91244
2. 73 156. 418 0. 40007 0. 91647
2. 74 156. 990 0. 39089 0. 92043
2. 75 157. 56.'I o. 38167 0. 92430
2. 76 158. 136 o. 37240 0. 92807
2. 77 1.58. 709 o. 36310 0. 93175
2. 78 159. 282 0. 35377 0. 93533
2. 79 159. 855 o. 34440 0. 93882
2.80 160. 428 o. 33499 0. 94222
2. 81 161. 001 0. 32555 0. 94553
2.82 161. 574 o. 31608 0.114873
2. 83 162. 147 0: 30658 0. 95184
2.84 162. 720 o. 29704 0. 95487
2. 85 163. 293 0. 28748 0. 95779
2. 86 163. 866 0. 27788 0. 96062
2.87 164. 439 0. 26827 0. 96335
2.88 165. 012 0. 25862 0. 96598
2. 89 165. 584 o. 24895 0. 96852
2. 90 166. 158 0. 23925 0. 97096
2. 91 166. 731 0. 22952 0. 97330
2. 92 167. 304 0. 21979 0. 97555
2. 93 167. 877 0. 21002 0. 97770
2. 94 168. 450 0. 20022 0. 97975
2. 95 169. 023 0.19042 0. 98170
2. 96 169. 596 0.18060 0. 98356
2. 97 170. 169 0. 17076 0. 98531
2.98 170. 741 0. 16089 0. 98697
2. 99 171. 314 0. 15101 0. 98853
3.00 171. 887 0. 14112 0. 98999
3.01 172. 460 0. 13121 0. 99135
3.02 173. 033 0.12129 0. 99262
3. 03 173. 606 0. 11136 0. 99378
3.04 174. 179 0. 10142  0. 99484
3. 05 I 174. 752 0. 09146 0. 99580
3.06 175. 325 0. 08150 0. 99667"
3. 07 175. 898 o: 07153 0. 99744
3. 08 176. 471 b. 06155 0. 99810
3. 09 177. 044 0. 05156 0. 99867
3.10 177. 617 0. 04i59 0. 99913
3. 11 178. 190 0.03159 0. 99950
3.12 178.·763 0. 02160 0. 99977
3. 13 179. 336 0. 01160 0. 99993
3. 14
I
179. 909 0. 00160 1. 00000
3.15 180. 482 0. 00841 0. 99997
L/j a
0 1. 0000
0. 5 1.0300
1. 00 1.1304
1. 05 1.1455
1.10 1. 1617
1. 15 1. 1792
1. 20 1.1979
l. 25 l. 2180
l. 30 l. 2396
l. 35 1.2628
1. 40 1. 2878 I
1. 45 1. 3146
1. 50 1. 3434
1: 55 1. 3744
T11ngent X
0. 60160
0. 58806
0. 57468
0. 56145
 0. 54837
~o. 53544
0. 52264
0. 50997
0. 49744
0. 48502
0. 47273
0. 46055
0. 44849
0. 43653
0. 42467
0. 41292
0. 40126
0. 38970
0. 37822
0. 36684
0. 35553
0. 34431
0. 33316
0. 32209
0. 31109
0. 30Q14
0. 28928
0. 27847
0. 26773
0. 25704
0. 24641
. 0. 23583
0. 22529
0. 21481
0. 20437
0.19397
0.18362
0.17330
0. 16301
0.15276
0. 14254
0. 13235
0. 12219
0. 11206
0. 10195
0. 09185
I
0. 08177
0. 07171
0. 06167
0. 05164
0.04162
0. 03161
0. 02160
0. Oll60
o: 00160
0. 00841
Xin SlneX Cosine X TangentX I radians
3. 16 0. 01841  0. 99983 0. 01841
3. 17 0. 02MO 0. 99960 0. 02841
3. 18 0. 03840  0. 99926 0. 03843
3. 19 0. 04839  0. 99883 0. 04845
3. 20 0. 05838 0. 99830 0.05848
3.21 0. 06836 0. 99766 0. 06852
3. 22  0: 01833 0. 99693 0. 07857
3. 23 0. 08829 0. 99609 0. 08864
3. 24 0. 09825 0. 99516 0. 09873
3. 25 ().10820 0. 99413 0. 10883
3. 26 0. 11814 I 0. 99300 0. 11896 3. 27 0. 12806 I
I
0. 99177 0. 12912
3. ~8 0.13797 0. 99044 0. 13930
3. 29 0. 14787 0. 98901 0. 14951 I
3. 30 0. 15774 0. 98748 0. 15975
3. 31 0. 16761 0. 98585 0. 17002
3. 32 0. 17746 0. 98412 0. 18033
3.33 1 0. 18729 0. 98230 o. 19067
3. 34  0. 19711 0. 98039 0. 2()105
3. 35 0. 20690  0. 97836 o. 21148
3. 36 0. 21668 0. 97624 0. 22195
3. 37 , 0. 22643 0. 97403 0. 23246
3. 38 0. 23616 0. 97172 0. 24303
3. 39 0. 24587  0. 96930 o. 25365
3.40 1 0. 25555 0. 96680 0. 26431
3. 41 0. 26520 0. 96419 0. 27504
3. 42 0. 27482  0. 96149 0. 28583
3. 43 0. 28443 0. 95870 0. 29668
3. 44 0. 29400 0. 95581 0. 30759
3. 45 0. 30354  0. 95282 0. 31857
3. 46 0. 31300  0. 94974 0. 32962
3. 47 0. 322M  0. 94656 0. 34074
3. 48 0. 33199 0. 94328 0. 35195
3. 49 0. 34141 0 .. 93992 0. 36322
3. 50 0. 35077 0. 93646 o. 37459
TABLE B
Tables of values of a, {3, and i' where
6(L/j cosec L/j1)
a= (L/j)2
fJ 3(1 ~ L /j cot L/j)
(L/j)2
3 (tan L /2j  L /2j)
(L/2j)3
A general relation existing between a, {3, and i' is
Table of "'• {3, and i' functions
Aa fJ A{J "Y Ay L/i
1.0000 1.0000 0
I. 0171 1.0256 0.5
l. 0737
I
1.1113 1.00
0. 0151 0. 0085 0. 0128
l. 0822 1.1241 1. 05
o. 0162 0. 0090 0.0138
1. 0912 l. 1379 1.10
0.0175 0. 0097 0. 0148
l. 1009 l. 1527 1. 15
0. 0187 0. 0105 0. 0159
1.1114 1.1686 l. 20
0. 0201 0. 0111 0. 0170
1.1225 1:.1856 l. 25
o. 0216 0. 0120 o. 0183
l. 1345 l. 2039 l. 30
0. 0232 0. 0128 0. 0196
1.1473 I l. 2235 l. 35
0. 0250 0. 0137 0. 0210
1.1610 1. 2445 1.40
0. 0268 0. 0147 0. 0226
1. 1757 l. 2671 1. 45
o. 0288 0.0158 0. 0243
1.1915 l. 2914 1. 50
o. 0310 0. 0169 0. 0260
1. 2084 1. 3174 1. 55
o. 0334 0. 0182 0. 0281
22
TABLE B.Continued
Table of a, {J, and 'Y functions'Continued .
L/J a Aa fJ AfJ 'Y A'Y L/J
1.00 1. 4078 1. 2266 1. 3455 1. 00
0. 0361 0. 0196 0. 0303
1. 65 1.4439 1. 2462 1. 3758 1. 65
0.0391 0. 0211 0.0327
1. 70 1. 4830 1. 2673 1. 4085 1. 70
0. 0422 0.0228 0. 0353
1. 76 1. 5252 1.2901 1. 4438 1. 75
0. 0458 0.11246 0.0383
1. 80 1. 5710 1. 3147 1. 4821 1. 80
0. 04118 0. 0267 o. 0416
1. 85 1. 6208 1.3414 1. 5237 l.85
0. 0.542 o. 0200 0. 0452
1. 90 1. 6750 1. 3704 1. 5689 1. 90
0. 0593 0. 0316 0. 0493
1. 95 1. 7343 1.4020 1. 6182 1. 95
0.0650 0.0345 0.0540
2. 00 1. 7993 1. 4365 1. 6722 2.00
0.0713 0.0377 0.0593
2.05 1. 8706 1.4742 1. 7315 2.05
0.0788 Q.0415 0. 0652
2.10 1. 9494 1. 5157 1. 7967 2. 10
0.0872 0. 0459 0. 0722
2.15 2.0366 1. 5616 1. 8689 2.15
0.0970 0. 0508 o. 0803
2. 20 2.1336 1. 6124 1.9492 2. 20
0. 1085 0. 0566 0. 0895
2. 25 2. 2421 1.6600 2. 0387 2. 25
0. 1220 00. 0635 0. 1005
2.30 2. 3641 1. 7325 2. 1392 2. 30
0. 1380 0. 07J6 O. ll37
2. 35 2. 5021 .1.8041 2. 2529 2. 35
0. 1574 o. 0813 0.1293
2. 40 2. 6595 1. 8854 2. 3822 2.40
0. <1172 0. 0450 0.0716
2. 425 2. 7467 1.9304 2. 4538 2.425
0. 0937 0. 0482 0. 0769
2.45 2. 8404 1.9786 2. 5307 2. 46
0. 1009 o. 0518 0. 0827
2. 475 2. 9413 2. 0304 2. 6134 2. 475
0. 1089 o. 0560 0. 0893
2.50 3. 0502 2. 0864 2. 7027 2. 50 o. uso 0. 0604 0. 0966
2. 625 3. 1682 2.1468 2. 7993 2. 525
0. 1282 0. 06.56 0. 1050 .2. 65 3. 2964 2. 2124 2. 9043 2. 55
0. 1397 0. 0714 O.U43
2. 575 3. 4361 2. 2838 3. 0186 2. 575
0. 1529 0. 0779 0. 1249
2. 00 3. 5890 2. 3617 3. 1435 2. 00
0.1680 0. 0856 0. 1372
2.625 3. 7570 2. 4473 3. 2807 2. 625
0.1852 0. 0942 0. 1513
2. 115 3. 9422 2. 5415 3. 4320 2.65
o. 2054 0.1043 0. 1678
2.676 4. H76 2. 6458 3. 5996 2.676
o. 2290 0. 1161 0. 1867
2. 70 4.3766 2. 7619 3. 7S63 2. 70
o. 2568 o. 1298 o. 2093
. 2. 725 4. 6334 2. 8917 3. 9956 2. 725
o. 2899 0. 1469 0. 2361
2. 75 4. 9233 3. 0386 4. 2317 2. 76
0.3297 0. 1666 0. 2685
2. 775 5. 2530 3. 2052 4. 5002 2. 775
0. 3785 0. 1912 0. 3080
2.80 5. 6315 3. 3004 4. 8082 2.80
0. 4387 o. 2210 0. 3568
2.825 6. 0702 3. 6174 5. 1650 2.825
0. 5163 0. 2600 0. 4202
2.85 6. 5865 3. 8774 5. 5852 2. l'5
0. 6094 0. 3065 0. 4948
2.875 7.1959 4.1839 6. 0800 2.875
o. 7384 o. 3711 o. 5998
2. 90 7. 9343 4. 5550 6. 6798 2. 90
0. 9096 0.4567 o. 7386
2. 925 8. 8439 5. 0117 7. 4184 2.9U
1.1476 o. 5758 0. 9319
2. 95 9. 9915 5. 5875 8.3503 2. 95
1.4930 0. 7484 1. 2113
2. 976 11. 4845 6. 3359 .9. 5616 2. 975
2. 0212 1. 0127 1. 6402
3.00 +13. 5057 +1. 3486 +11. 2013 3. 00
1.0238 0. 5127 0. 8304
3. 01 +14. 5295 +1. 8613 +12. 0317 3. 01
1. 1924 0. 5970 0. 9671
3. 02 +15. 1219 + 8. 4584 +12. 9988 3.02
I. 4063 0. 7040 1.1405
3.03 +11.1282 +9.1623 +14. 1393 3.03.
1. 6834 o. 8425 I. 3651
3. 04 +18.8117 +10. 0049 +15. 5044 3. 04
2. 0513 1. 0265 1. 6633
3. 05 +20. 8629 +11. 0314 +17. 1677 3. 05
2. 5547 l . 2782 2. 0711
3. 06 +23. 4176 +12.3096 +rn. 2388 3. 06
3.2684 1.6350 2. 6498
3. 07 +26.6800 +13. 9446 +21. 8886 3.07
4. 3301 2. 1659 3. 5103
23
TABLE BContinued
Table of a, {3, and 'Y /unctionsContinued
L/j a II.a fJ 11.{J '( 11.y L/J
3.08 +31. 0160 +16.1105 +25. 3989 3. 08
6.0084 3.0051 4.8712
3. 09 +37.0244 +19.1156 +ao. 2101 3.09
8.8989 4. 4503 7. 2137
3.10 +45. 9234 +23. 5659 +37.4839 3.10
14. 5332 7. 2675 11. 7808
3. 11 +60.4566 +ao. 8334 +49.2M7 3.11
27. 9956 13. 9987 22. 6930
3. 12 +88.4522 +44. 8321 +7L9577 3.12
76. 2965 38. 1491 61. 8440
3, 13 +164. 7487 +82. 9812 +133. 8017 3.13
1034. 4142 517. 2088 838. 4545
3.14 +1199. 1629 +600.1900 +972. 2562 3.14
co co co
3. 15 227.1668 112. 9747 183. 8716 3. 15
123. 4092 61. 7065 100.0325
3.16 103. 7576 51. 2692 83. 8391 3.16
36. 5229 18. 2614 29.6049
3. 17 67.2348 33. 0068 54.2342 3.17
17. 5035 8. 7527 14. 1885
3.18 49. 7313 24. 2542 40.0458 3.18
10. 2712 5. 1365 8. 3263
3.19 39. 4600 19. 1176 31. 7195 3.19
6. 7537 3. 3778 5. 4750
3.20 32. 7063 15. 7398 26. 2445 3.20
4. 7788 2. 3903 3.8742
3. 21 27.9276 13.3495 22. 3703 3. 21
3. 5693 1. 7807 2. 8858
3.22 24.3683 11. 5688 19. 4845 3. 22
2. 7541 1. 3779 2. 2330
3.23 21. 6142 10.1909 17. 2515 3. 23
2. 1940 1.0980 1. 7790
3.24 19. 4202 9.0929 15. 4725 3.24
1. '1890 0.8955 1.4508
3.25 17. 6312 8.1975 14. 0218 a. 25
1. 4866 0. 7443 1. 2057
3. 26 16. 1447 7.4532 12. 8161 3.26
1.2548 0. 6284 1.0178
3.27 14.8899 6.8248 11. 7983 3. 27
1. 0733 0. 5376 0.8707
3. 28 13. 8166 6. 2872 10. 9276 3. 28
0. 9285 0. 4652 0. 7633
3. 29 12.8881 5.8219 10. 1743 3. 29
0. 8111 0. 4066 0. 6581
3.30 12.0770 5. 4154 9. 5162 3. 30
4. 6521 2. 3366 3. 7784
3.40 7. 4248 3.0787 5. 7378 3.40
2. 04711 1. 0354 1. 6681
3. t50 5.3769 2.0433 4. 0697 3. t50
1.1477 0. 5861 o. 9389
3. 60 4. 2292 1.4572 3.1308 3. 60
0. '302 0. 3785 0.6016
3. 70 ~a.4990 1. 0787 2. 5292 3. 70
0. 5029 0. 2659 0. 4179
3.80 2. 9961 0. 8128 2. 1113 3.80
0. 3647 0.1981 0.3070
3.90 2. 6314 0.6147 1. 8043 3.90
0. 2744 0. 1544 0. 2349
4.00 2. 3570 0. 4603 1. 5694 4. 00
0. 2116 0. 1248 0.1854
4. 10 2. 1454 0.3355 1. 3840 4. 10
0. 1662 0. 1038 0. 1498
4. 20 1. 9792 0. 2317 1. 2342 4.20
0. 1317 0. 0887 0. 1237
4. 30 1. 8475 0.1430 1.1105 4. 30
0.1046 0.0778 0. 1036
4.40 1. 7429 0. 0652 1.0069 4.40
0.0826 0.0696 0. 0881
4.50 1.6603 f0. 0044 0. 9188 4.50
0.0641 0.0638 0.0757
4.60 1. 5962 +o. 0682 0. 8431 4.60
0.0810 0.1169 0.1235
4.80 1. 5152 +o.1851 0. 7196 4. 80
0. 0238 0. 1124 0.0962
5.00 1.4914 +o. 2975 0.6234 5.00
0. O!i68 0.1520 0. 0938
5.25 1. 5842 +o. 4495 0. 5296 5.25
0.1964 0. 1975 0. 0733
5.5  1. 7446 +o.6470 0.4563 5. 5
0.4898 0. 3277 0. 0589
5.75 2. 2344 +0.9747 0.3974 5. 75
1. 5111 0:8268 0.0482
6.0 • 3. 7456 +t.8015 0. 3492 6.0
25. 3412 12. 7331 0.0404
6.25 29.0867 +14.5346 0.3088 6.25
co co 0. 0048
2r =F<;O :f:co 0.3040 2r
co co 0. 0295
6.5 +4.1490 2.0242 0.2745 6.6
24
TABUE c
Values of ah, flh, and 'Yb for use in the precise threemoment
equation for beams subjected to a uniformly
distributed load and on axial tension:
It will be noted that neither 7r/2 nor 7r is a critical
point for values of ah, flh, or 'Yb·
These values are derived directly fro:!Il the tables
compiled by Mr. Arthur Berry in his paper, "The
Calculation of Stresses in Airplane Wing Spars, " published
in the Transactions of the Royal Aeronautical
Society, 1919, the argument X in these tables being
twice the argument II used by Berry.
L/J ah . .<\ah /3• 4/3h °)'h . .<\°)'b L/j
   · 
0.00 1. 0000 1.0000 1. 0000 0. 00 o. 0284 0. 0163 o. 0244
0. 50 0. 9716 0. 9837 0. 9756 0. 50 o. 0771 0. 0446 0. 0664
1.00 0. 8945 0. 9391 o. 9092 1.00
0. 0097 0. 0057 0. 0083
1.05 0. 8848 o. 9334 0. 9009 1. 05 o. 0100 0. 0058 o. 0087
1.10 0. 8748 o. 9276 0. 8922 1.10
0. 0100 0. 0060 0. 0089
1. 15 0.8647 0. 9216 0.8833 1. 15
0. 0105 0. 0061 o. 0090
1.20 0. 8542 0. 9155 0.8743 I. 20
0. 0106 0. 0062 0. 0092
I. 25 0.8436 0. 9093 o. 8651 1.25
0. 0108 0. 0065 0. 0094
1.30 0.8328 o. 9028 o. 8557 1. 30
0. OHO 0. 0065 0. 0096
I. 35 0. 8218 o. 8963 0.8461 I. 35
0. Olli 0. 0066 0. 0097
1.40 0.8107 0.8897 0. 8364 I. 40
0. Oll3 0.0067 o. 0098
1. 45 0. 7994 0.8830 0. 8266 1. 45
0. Oll3 0. 0068 0.0099
0
TABLE CContinued
I L/j ab. _ .6ah /3b. 4/3• "Yb ~1~
1. 50 o. 7881 0. 8762 0. 8167 1.50
0. Oll4 0. 0068 o. 0100
I.55 0. 7767 0. 8694 0.8067 1. 55
0. Oll5 0. 0069 o. 0100
1.60 o. 7652 0.8625 0. 7967 l . 60
0. 0115 0. 0070 0. 0100
1.65 0. 7537 0. 8555' 0. 7867 1. 65
0. Oll6 0. 0070 0. 0101
1. 70 0. 7421 o. 8485 o. 7766 1. 70
0. Oll6 0. 0070 0. 0102
1. 75 0. 7305 0. 8415 0. 7664 I. 75
0. Oll6 0. 0071 0. 0104
1. 80 o. 7189 0.8344 o. 7560 1.80
0. Oll6 0. 0071 o. 0103
1. 85 o. 7073 0. 8273 0. 7457 1. 85
0. Oll5 o. 0071 o. 0102
1. 90 o. 6958 o. 8202 0. 7355 1. 90
0. Oll5 o. 0071 0. 0102
1. 95 o. 6843 o. 8131 0. 7253 1.95
0. Oll5 0. 0071 0.0101
2. 00 0. 6728 0.8060 o. 7152 2.00
0. Oll4 0. 0071 0. 0101
2.05 o. 6614 o. 7989 0. 7051 2. 05
0. Oll3 0. 0071 0. 0101
2.10 o. 6501 0. 7918 0. 6950 2. 10
0. 0112 0. 0071 0. 0100
2. 15 o. 6389 0. 7847 0. 6850 2. 15
0. Olli 0. 0070 0. 0100
2. 20 0. 6278 0. 7777 0. 6750 2. 20
0. 0111 o. 0070 o. 0098
2. 25 0.6167 0. 7707 0. 6652 2.25
0.0109 0. 0070 0.0097
2.30 0. 6058 o. 7637 0. 6553 2. 30
0. 0108 o. 0069 o. 0098
2. 35 0. 5950 o. 7568 o. 6457 2. 35
0. 0107 o. 0069 o. 0097
2.40 o. 5843 0. 7499 0. 6360 2.40
0.0106 0. 0069 0. 0095
2.45 o. 5737 0. 7430 o. 6265 2. 45 o. 0104 0. 0068 o. 0095
2.50 0. 5633 0. 7362 0. 6170 2. 50
0. 0103 0. 0067 o. 0093
2.55 0. 5530 0. 7295 0. 6077 2. 55
0. 0101 o. 0067 0. 0092
2. 60 0. 5429 o. 7228 0. 5985 2. 60
0. 0100 o. 0066 0. 0092
2.65 0. 5329 o. 7162 0. 5893 2. 65
0.0099 0. 0065 o. 0090
2. 70 o. 5230 0. 7097 0. 5803 2. 70
0. 0097
iJ. 7032
0. 0065 0. 0088
2. 75 0. 5133 0. 5715 2. 75
0. 0096 0. 0065 o. 0088
2.80 0. 5037 0. 6967 0. 5627 2.80
0. 0094 o. 0064 o. 0085
2. 85 o. 4943 o. 6903 0. 5542 2. 85
0. 0092 0. 0063 0. 0085
2.90 0. 4851 0. 6840 0. 5457 2. 90
0.0091 0. 0062 0. 0085
2. 95 o. 4760 0. 6778 (J.5372 2. 95
0. 0090 0. 0062 0. 0084
3. 00 o. 4670 o. 6716 0. 5288 3. 00
0. 0087 0. 0061 o. 0083
3. 05 0, 4583 0. 6655 0. 5205 3. 05
0. 0087 o. 006tJ o. 0080
3. 10 0. 4496 0. 6595 0. 5125 3. 10
0. 0085 o. 0059 0. 0080
3.15 0. 4411 o. 6536 0. 5045 3. 15
0. 0083 0. 0060 0. 0077
3. 20 .o. 4328 0. 6476 0. 4968 3. 20
.....